the dynamics of keynesian monetary growth-macrofoundations-chiarella & flaschel

MMMMThis page intentionally left blank

This book is in the tradition of non-market clearing approaches to macrodynamicanalysis. It builds a series of integrated disequilibrium growth models of increasingcomplexity, which display the economic interaction between households, firms andgovernment across labor, goods, money, bonds, and equitiesmarkets. Chiarella andFlaschel demonstrate how macrodynamics can be developed in a hierarchical wayfrom economically simple structures to more advanced ones. In addition theyinvestigate complex macrodynamic feedback mechanisms.The book is organized into seven chapters. Chapter 1 discusses traditional

macrodynamic model building. Chapters 2—4 show how Keynesian disequilibriumgrowth can be obtained from Tobin and Keynes—Wicksell monetary growthmodels. Chapter 5 treats the cases of substitution in production, and chapter 6provides the workingmodel of the book. Chapter 7 discusses further extensions andgives an outlook on future work.

is Professor of Finance at theUniversity of Technology,Sydney.He holds doctorates in applied mathematics and in economics from the Universityof New South Wales. His main research interests are economic dynamics andquantitative finance. His work appears in journals such as JEDC, JEBO,Macro-economic Dynamics, Economic Modelling, Applied Mathematical Finance and theJournal of Computational Finance. He is the author of Elements of a NonlinearTheory of Economic Dynamics and the co-author, with Peter Flaschel, ofDisequilib-rium, Growth and Labor Market Dynamics.

is Professor of Economics at the University of Bielefeld. Hecompleted his doctoral thesis in mathematics at the University of Bonn and hishabilitation thesis in economics at the Free University of Berlin. He publishesextensively on economic theory and macroeconomic dynamics in journals such asEconometrica, JEBO, JEDC, Macroeconomic Dynamics, Economic Modelling andThe Manchester School. He is the author of Macrodynamics, Dynamic Macro-economics and the co-author, with Carl Chiarella, of Disequilibrium, Growth andLabor Market Dynamics.

The Dynamics of Keynesian Monetary Growth

The Dynamics ofKeynesian MonetaryGrowth:

Macro Foundations

CARL CHIARELLASchool of Finance and EconomicsUniversity of Technology, SydneyAustralia

PETER FLASCHELDepartment of EconomicsUniversity of BielefeldGermany

The Pitt Building, Trumpington Street, Cambridge, United Kingdom

The Edinburgh Building, Cambridge CB2 2RU, UK40 West 20th Street, New York, NY 10011-4211, USA477 Williamstown Road, Port Melbourne, VIC 3207, AustraliaRuiz de Alarcón 13, 28014 Madrid, SpainDock House, The Waterfront, Cape Town 8001, South Africa

http://www.cambridge.org

First published in printed format

ISBN 0-521-64351-1 hardbackISBN 0-511-03668-X eBook

Carl Chiarella and Peter Flaschel 2004

2000

(Adobe Reader)

©

Contents

List of figures page xForeword by Richard H. Day xvPreface xviiiAcknowledgments xxNotation xxii

General introduction 1

1 Traditional monetary growth dynamics 101.1 Introduction 101.2 Macro foundations of macroeconomics 121.3 Basic Tobin models of monetary growth 241.4 Basic Keynes-Wicksell models of monetary growth 311.5 Basic AS—AD growth models 391.6 The modeling of expectations 461.7 A new integrated approach to Keynesian monetary

growth 611.8 Mathematical tools 65Appendix 67

2 Tobinian monetary growth: the (neo)Classicalpoint of departure 692.1 The basic equilibrium version of Tobin’s model

of monetary growth: superneutrality and stability? 712.2 The money-market disequilibrium extension:

further stability analysis 822.3 Labor-market disequilibrium and cyclical monetary

growth 922.4 General equilibrium with a bond market: concepts of

disposable income and Ricardian equivalence 102

vii

2.5 A general disequilibrium version of the neoclassicalmodel of monetary growth 112

2.6 Outlook: independent investment behavior andWicksellian price dynamics 123

3 Keynes—Wicksell models of monetary growth:synthesizing Keynes into the Classics 1273.1 The general prototype model 1293.2 The intensive form of the model 1363.3 The Goodwin growth cycle case 1403.4 The Rose employment cycle extension 1463.5 Monetary growth cycles: the basic case 1543.6 Expectations and the pure monetary cycle 1593.7 The real and the monetary cycle in interaction 1683.8 Outlook: less than full capacity growth 171

4 Keynesian monetary growth: the missing prototype 1734.1 A general Keynesian model of monetary growth 1754.2 Comparative statics: the IS—LM subsector 1844.3 Growth cycle implications 1904.4 Employment cycle extensions 1994.5 Keynesian monetary growth: the basic case 2074.6 Monetary and real factors in Keynesian cyclical

growth dynamics 2144.7 Outlook: adding smooth factor substitution 220Appendix 1: The Benassy business cycle model 231Appendix 2: Technical change, wage taxation, averageinflation and p-star expectations 235

5 Smooth factor substitution: a secondary andconfused issue 2425.1 The Tobin case: one further integrated law of motion 2435.2 The Keynes—Wicksell case: increased stability through

increased flexibility 2535.3 The Keynesian case with smooth factor substitution 2595.4 Outlook: sluggish price as well as quantity dynamics 275

6 Keynesian monetary growth: the working model 2786.1 Introduction 2786.2 The Kaldor—Tobin model of monetary growth 2836.3 An integrated Keynes—Metzler model of monetary

growth 293

viii Contents

6.4 A (5� 1)-D modification of the six-dimensionalKeynes—Metzler model 314

6.5 Outlook: macroeconometric model building 335

7 The road ahead 3407.1 Endogenous long-run growth and employment 3417.2 The dynamic structure of the model 3487.3 Analysis of the employment subdynamics 3507.4 Analysis of the growth subdynamics 3547.5 Analysis of the complete dynamical system 3567.6 Some numerical simulations 3587.7 Summary and directions for future research 372

References 383Author index 394Subject index 397

ixContents

Figures

1.1 Phase diagram of the dynamics under adaptiveexpectations page 51

1.2 Instability in the perfect foresight limit 511.3 Jump to linearization of stable manifold 531.4 The true and the perceived system 541.5 Nonlinearity in the money demand function 551.6 Relaxation oscillation in inflationary expectations 571.7 Time series presentation of the relaxation oscillation 582.1 Simple nonlinear money demand function 892.2 Bounded fluctuations for disequilibrium monetary growth 902.3 The case of relaxation oscillations or limit limit cycles 912.4 Disentangled real cycle in the Tobin model 1222.5 Disentangled monetary cycle in the Tobin model 1232.6 Combined real and monetary cycle of the Tobin model 1242.7 Combined real and monetary cycle of the Tobin model

with additional nonlinearity in the price reaction function 1253.1 Ceilings to the validity of the Goodwin growth cycle

approach 1453.2 A nonlinear law of demand in the labor market 1493.3 Implications of nonlinearity in the labor market 1503.4 (a) A nonlinear investment-savings relationship; (b) a Rose

limit cycle in the fixed proportions case 1513.5 The real cycle of the Keynes—Wicksell model 1543.6 The two Routh—Hurwitz coefficients a

�, b 158

3.7 Phase diagram of the pure monetary cycle 1673.8 Simulation of the pure monetary limit cycle 1683.9 Coupled real and monetary oscillators 1704.1 The denominator in the effective demand function (4.36) 1874.2a Effective demand: a too weak capacity effect or s

�� i

�188

4.2b Effective demand: a strong capacity effect 189

x

4.3a Case 1: the ‘‘paradise’’ case 1924.3b Case 2: the ‘‘orthodox’’ case 1924.3c Case 3: the ‘‘mixed’’ case 1934.4 The parameter �

�(V) of the wage adjustment function

��(V) (V� 1) 194

4.5 A region of global stability for case 2 1954.6 Instability for case 2 via the Rose effect 2024.7 Stability for case 3 via the Rose effect 2034.8 The nonlinear Phillips-curve mechanism once again 2034.9 Viability in the locally unstable case 2 (the real cycle,

case 1) 2054.10 A second Phillips-curve mechanism 2054.11 Viability in the locally unstable case 3 (the real cycle,

case 2) 2074.12 The stability switch in case 1 2084.13 Determination of the bifurcation parameter value ��

�213

4.14 The pure monetary cycle 2194.15 A numerical example for the pure monetary cycle 2204.16 The nonlinear component of the investment function 2224.17 A nonlinear goods-market equilibrium curve 2234.18 The phase diagram of a pure real cycle 2244.19 A simulation of the pure real cycle 2254.20a A simulation of the joint monetary and the real cycle in

the intrinsically nonlinear case (with no investmentnonlinearity) 226

4.20b A simulation of the joint monetary and the real cycle inthe extrinsically nonlinear case 228

4.21 Benassy’s money wage Phillips—curve 2334.22 Constructing a viability domain for the Benassy model 2345.1 The determination of the Hopf-bifurcation parameter 2495.2 The non-superneutrality of money 2525.3 A restricted neoclassical production function 2565.4 The viability domain of the Rose dynamics under smooth

factor substitution 2575.5 Potential and actual employment and output 2646.1 Hopf bifurcation curves, stable limit cycles (projections),

or stable corridors 3036.2 Hopf bifurcation loci of the inventory cycle for Z� 0 3106.3 Hopf bifurcation curves, stable limit cycles, and stability

corridors for Z� 0 3126.4 Six-dimensional bifurcation loci and a limit cycle for h

�� 0.2 (Q� 0) 321

xiList of figures

6.5 Six-dimensional Bifurcation-loci and a limit cycle for h�

� 0.8 (Q� 0) 3236.6 A period-doubling route to complex dynamics 3246.7 At the edge of mathematical boundedness 3256.8 No steady-state inflation 3296.9 Steady-state inflation and period 1 limit cycles 3306.10 Steady-state inflation and period 4 limit cycles 3306.11 Steady-state inflation and period 16 limit cycles 3316.12 Steady-state inflation and complex dynamics 3316.13 A bifurcation diagram for the dynamics considered in

figures 6.9—6.12 3336.14 The largest Liapunov exponent of the dynamics considered

in figure 6.13 3336.15 A test for sensitivity with respect to initial conditions for

the above-shown attractor 3347.1 Phase plots and times series representations over a time

horizon of 200 years (6D case) 3607.2 Phase plots and times series representations over a time

horizon of 1,000 years (6D case) 3617.3 Bifurcation diagram of the 6D case for �

�� [0.5, 20] 362

7.4 Downwardly rigid money wages at the inflationless steadystate 363

7.5 Phase plots and times series representations of endogenous‘‘animal spirits’’ over a time horizon of 220 years (7D case) 364

7.6 Phase plots and times series representations of endogenous‘‘animal spirits’’ over a time horizon of 1,000 years(7D case) 365

7.7 Phase plots and times series representations of endogenous‘‘natural growth’’ over a time horizon of 220 years(8D case) 366

7.8 Phase plots and times series representations of endogenous‘‘natural growth’’ after a transient period of 1,000 years(8D case) 367

7.9 Bifurcation diagram for �� [0.5, 5] in the case of

endogenous growth (8D case) 3687.10 High adjustment speeds of wages and the occurrence of

‘‘complex’’ dynamics (8D case) 3697.11 Phase plots and times series representations of an

endogenous determination of the NAIRU-based rate V�(7D case) 370

xii List of figures

7.12 Phase plots and times series representations of anendogenous determination of the NAIRU-based rate ofemployment V� (7D case) 371

7.13 Phase plots and times series representations of anendogenous determination of ‘‘natural’’ rates ofemployment and of growth (9D case) 372

7.14 Phase plots and times series representations of smallfluctuations in the level of economic activity (9D case) 373

7.15 Phase plots and times series representations of anendogenous determination of ‘‘natural’’ rates ofemployment and of growth (9D case) without extrinsicnonlinearities 374

xiiiList of figures

Foreword

Richard H. Day

In his effort to reorient economic theory so that it might offer an explana-tion of severe and prolonged recessions and insights concerning the possi-bilities and limitations of fiscal and monetary policies for dealing withthem, Keynes introduced two factually based assumptions: first, price andwage stickiness; second, independently determined savings and investmentvariables. In developing the implications of these two facts, Keynes ex-ploited the concept of demand, not at the usual level of the market for asingle good, but at the level of the entire economy for the aggregate of allgoods. Thus, for example, instead of an Engle curve for a single good, whichgives the demand for a good in terms of real income, he exploited thedependence of aggregate demand for all goods on income, that is, theconsumption function. With the real money rate of interest as the onlyendogenously determined price — in this case the price (or opportunity cost)of using money as an idle balance — the money market is seen to play apotential role; potential, because its role depends on sensitivity to interestrates on the markets for goods and money.Although his analytical derivations were static and focused on a new

kind of persistent unemployment situation, Keynes had in mind a dynamictheory.He fully intended to illuminate the tendency of themarket economyto fluctuate due to the interactions between the monetary and real goodssectors.Keynes’ ideas were obviously relevant.Within a decade they led to a new

field of economics based on a reduction of the microeconomics of manygoods and prices to amacroeconomics based on ameasure of the aggregateof all goods andmoney. Extreme assumptions were necessary to reduce thetheory to the graphical dimensions required by contemporary pedagogy.This yielded the standard by which the theory became widely known:Hicks’ ingenious IS—LM framework.The deficiencies of this static, simplified version were obvious. Instead of

prices and wages that adjusted stickily, it assumed prices that did not

xv

adjust at all; it included investment but not capital accumulation; it treatedthe money supply as exogenous instead of incorporating a dependence oncredit conditions and government finance. These deficiencies motivated alarge body of work aimed at reducing these deficiencies, work that con-tinues to the present as readers of this volume will come to appreciate. But,in the meantime, a quite different body of work veered off this (then)mainstream approach. Instead of building on aggregate supply and de-mand of heterogeneous firms and households out of equilibrium, it built onthe concept of a Robinson Crusoe or a representative agent in intertem-poral equilibrium. Since only a single agent is modeled, there is no problemof coordination among markets, no need to consider savings and invest-ment out of equilibrium, and no need to consider fluctuations caused bythe interaction of money markets and goods. Instead, the source of fluctu-ation is sought in terms of unexplained, exogenous shocks which pushequilibria around.The intellectual advantage of this approach lies in its reliance on the

equilibriumassumptionwhich is embodied in the principle of optimality. Itenables the derivation of ‘‘optimal trajectories’’ for consumption, labor,and capital. By means of an extended duality principle, the supportingcompetitive equilibrium price trajectories are implied. Along such equilib-rium paths involuntary unemployment and excess capacity do not occur.This is not the occasion to address in detail the relationship between

these two approaches except to emphasize that the dynamic, aggregatesupply/demand approach represents the economy as one that adjusts outof perfect coordination to disequilibrium signals, in contrast to the equilib-rium approach which represents the economy as a perfectly coordinatedprocess with no need for mechanisms of adaptation to deal with discrepan-cies among the constituent parts.As the American academic establishment expanded during the last

half-century into a new kind of mass market for education and science, itbegan to exhibit a herding phenomenon not unlike fashions in consumergoods. For a time, the macroeconomic fashion leaders were centered at aHarvard/MIT/Penn nexus in the persons of Hansen, Duessenberry, Sam-uelson, and Klein. Out-of-equilibrium thinking ruled macroeconomic the-ory and econometrics until the mid 1970s. When the leading macro-economic equilibrium pundit moved from Carnegie Mellon to Chicago, anew fashion of equilibrium macroeconomics emerged with a new center ofgravity. The new fashion leaders spread in due course to Harvard andStanford and many points in between and beyond.In the meantime, the serious work of extending the out-of-equilibrium

aspect of macroeconomics so as to remove its deficiencies, so as to improveits ability to explain real world events and so as to improve its potential for

xvi Foreword

policy repercussion analysis has continued. In England and especially inGermany and Italy, as well as to a lesser extent in the USA, this stream ofwork has continued until a more general and more satisfactory theory hasemerged. Its potential for illuminating macroeconomic phenomena hasbeen enhanced and its potential for providing new understanding of fiscaland monetary policy improved.This book by Chiarella and Flaschel is a contribution to this out-of-

equilibrium stream of macroeconomic theory. Beginning with Tobin’smonetary growth analysis, it successively introduces realistic, complicatingrelationships that eliminate, step by step, some of the major deficiencies inthe earlier Keynesianmodels. It gives a meticulous analysis of eachmodel’sproperties and an equally meticulous explanation of each model’s relation-ship to the contributions of other scholars. Anyone who wants to under-stand the development of macroeconomic thinking as a whole and whowants to see the modern development of the out-of-equilibrium approach,will want to study this volume.The authors dramatically demonstrate the power of the dynamic point

of view, and the potential for explaining apparent anomalies by en-dogenous economic forces. For example, the scatter of data suggesting arelationship between the rate of price changes and unemployment hasusually been explained in terms of shifting Phillips curves. However, whenthe scattered dots are connected in a time sequence, irregular Phillipsspirals are revealed. To theorists of dynamics, such spirals suggest anunderlying endogenous mechanism, not stationary points at the intersec-tion of exogenously shifting curves. By chapter 4 of the present volume,somewhat similar spirals are shown to emerge from endogenous, out-of-equilibrium, real/monetary interactions, a finding of great potential im-portance.The authors modestly present their findings as work in progress, and so

it is, but it is, nonetheless, a work of consummate scholarship. I have beenfortunate in having been able to follow the gradual accumulation of theauthors’ and their collaborators’ studies to their present state. It is anappropriate stage to present it in this integrated form. Every seriousstudent of macroeconomic theory will want to know what they have done,for in this work they will find a comprehensive analytical exegesis of thesteps by which the theory has reached its present state at the frontier, andan excellent jumping off point for further research. It seems to me likelythat it is only a matter of time before empirical studies, based on models ofthe kind analyzed here, will achieve a new breakthrough in understandingreal economic data and a new basis for predicting policy analysis.

xviiForeword

Preface

‘‘Macroeconomics has never reached a consensus and probably never will. Thesubject is too diverse and the approaches too varied for that to become likely.’’(S. Turnovsky, Methods of Macroeconomics Dynamics)

This book provides the reader with a systematic study of macrodynamicmodels of monetary growth in the Tobin, the Keynes—Wicksell and theKeynesian (if it exists) tradition. Our point of departure is, therefore, thecore of descriptive macrodynamicmodels of monetary growth of primarilytraditional origin; recent contributions of neo- or post-Keynesian type aswell as other schools of thought are given scant consideration in this book.Instead, we considerably extend and refine the aforementioned modeltypes so that they give rise to a hierarchical sequence of fully integratedmacrodynamic models, each providing an improvement on the shortcom-ings of one or more structural equations of its predecessor. In this way wearrive at the formulation of an integrated model of the Keynes—Metzlertype, with both sluggish price and quantity adjustment and under- oroveremployment of both labor and capital, whichmay be considered as theworking Keynesian prototype model of IS—LM growth.Yet, this model type also has its shortcomings, so the hierarchical

structure proposed here does not end with it, but rather will be continuedin future research by way of more refined treatments of asset markets, ofexpectations, of the role of income distribution, of international trade ingoods and financial assets, of stochastic influences, and so on. In this regardwe view this book as providing ‘‘macro foundations,’’ or a systematic wayto proceed from elementary studies of models of cycles and growth with afull range of markets (labor, goods, money, bonds, equities) to ever morerefined and detailed ones. Partial ‘‘micro foundations’’ for the structuresexist in the literature and, of course, must be improved as well, but this hasto be done based on the knowledge of what indeed has to be micro

xviii

founded, and thus put on a firmer basis.This book is the product of a continuing collaboration between the

authors which began in 1992 when the first author was on study leave atthe University of Mannheim in Germany. At a number of meetings duringthat period we realized that we shared a strong, common desire to set up aframework within which the non-market-clearing approach to dynamicmacroeconomics could be built in a systematic, consistent, and transparentmanner, starting from mainstream contributions to disequilibrium growthdeveloped in the sixties, seventies, and eighties. We have sought to con-struct a framework in which such mainstream contributions to the non-market-clearing paradigm could be reformulated on a common basis andextended systematically, leading successively to more and more coherentintegratedmodels of disequilibrium growth with progressively richer inter-actions between markets and sectors. In this way, we sought a frameworkto which further refinements, in terms of more markets, more agents, moreadvanced behavior of agents, could be added or inserted in a natural way,far beyond even the general working model of traditional Keynesianmonetary growth that is the focus of this book. Indeed, in other work wehave already started the task of these further extensions in several direc-tions, and these are alluded to in the final chapter. Of course, we must leaveto the reader to judge whether we have succeeded in our aims of providingwhat we would call macro foundations of traditional macrodynamics onthe basis of which more recent contributions to the non-market-clearingapproach to economic dynamics may be reconsidered, evaluated and usedas macro perspectives for the project we have begun with this book.

xixPreface

Acknowledgments

Thework has progressed thanks to almost annual visits since 1993 of PeterFlaschel to the School of Finance and Economics at the University ofTechnology, Sydney and almost equally frequent visits byCarl Chiarella tothe Faculty of Economics at the University of Bielefeld.We are both deeplyindebted to our respective institutions for the very strong financial supportwe have received which made these various visits possible, as well asother infrastructure support which allowed this project to be brought tocompletion.A number of professional colleagues deserve special thanks. In particular

Willi Semmler, who has offered constant encouragement and supportthroughout this project and the other related projects of the authors whichare discussed in the final chapter. Richard Day, Reiner Franke, GangolfGroh, Christian Groth, Cars Hommes, Klaus Jaeger, Reinhard John,Ingrid Kubin, Thomas Lux, Hans-Walter Lorenz, Reinhard Neck, Mat-thias Raith, Hans-Jurgen Ramser, Rajiv Sethi, and Peter Skott offeredvaluable comments as discussants at presentations of aspects of the ma-terial of this book at various international conferences and on otheroccasions. Of course, none of the aforementioned is responsible for theremaining errors in this work, neither with respect to form nor with respectto substance.We owe a particular debt of gratitude to Alexander Khomin, formerly of

the School of Finance and Economics at the University of Technology,Sydney, and now at the Commonwealth Trading Bank, Australia. Hedesigned and built the C�� computer packagewhich we used to performmany of the simulations of the model reported both here and in our otherpublished work.We are indebted to the anonymous referees who read the original

version of the manuscript and offered many suggestions for its improve-ment. We would also like to thank Ashwin Rattan of Cambridge Univer-

xx

sity Press for all that he has done to make the publication process aspainless as possible.Finally, we would like to express our thanks to the two persons who

have borne the biggest cost during the preparation of this book, our wivesLynette Siew-Hon Chiarella and Sigrid Luchtenberg.

xxiAcknowledgments

Notation

The notation employed throughout this book is subdivided into staticallyor dynamically endogenous variables and parameters, a subdivision whichis here presented from the perspective of chapter 7:�

A Statically or dynamically endogenous variables�Y�S�S

��S

��S

�Potential output

Y Output (�Y��potential output in general)Y��,Y�

��Disposable income (index c: of capitalists, index e:perceived)

Y Aggregate demand C� I� �K�GY Expected aggregate demandL�,L Employed workforce, employment of the employed

workforce (L�L� with the exception of chapter 7)C ConsumptionI InvestmentI� Planned investment I�I (I�� I�N� actual

investment)r Nominal rate of interest (price of bonds p

� 1)

p

Price of equitiesS�

Private savingsS�

Firms’ savingsS�

Government savingsS�S

��S

��S

�Total savings

T Real taxes (T�,T

�of workers and capitalists)

G Government expenditure

� The NAIRU-employment rate — denoted by V� in the following is the Non-Accelerating-Inflation-Rate-of-Utilization (here of the labor force), i.e., the employment-complement ofthe NAIRU of the literature. Starting with chapter 4 we shall make use in addition of aNAIRU conceptU� with respect to the rate of capacity utilizationU of the capital stockK.

� Some of these variables will be given parameters in the earlier chapters of this book.

xxii

� Rate of profit (expected rate of profit �)V�L�/L Rate of employment (V� the NAIRU employment rate)V��L/L� Utilization rate of the employedU�Y/Y� Rate of capacity utilization (U� the NAIRU rate of

capacity utilization)K Capital stockw Nominal wagesp Price levelp* p-star price level of the FED/German Bundesbankv� Velocity of money circulation� Expected rate of inflation (medium run average)M Money supply (index d: demand, growth rate

�)

L Normal labor supplyB Bonds (index d: demand)E Equities (index d: demand)W Real wealthN Stock of inventoriesN Desired stock of inventoriesI Desired rate of inventory change Trend growth rate of the capital stockn� n

�Natural growth rate

n�

Rate of Harrod neutral technical change��N/K Inventory—capital ratio� Real wage (u��/x the wage share)u��/x Wage share (x is labor productivity, see below)y�Y/K Output—capital ratio

B Parameters (all parameters represent positive scalars)y� Potential output—capital ratiox Output—labor ratio (labor productivity)� Depreciation plus inventory ratei, i

�, i�

Investment parametersh�, h

�Money demand parameters

�

Steady growth rate of money supply��,��,��

Wage adjustment speed parameters��

Price adjustment speed parameter��, �� Inflationary expectations adjustment speed

parameters��

Rate of employment adjustment parameter�v� NAIRU adjustment parameter�� Trend growth adjustment parameter��

Demand adjustment parameter

xxiiiNotation

�� Desired inventory output ratio

��

Inventory or natural growth rate adjustmentparameter

�� Demand expectations adjustment parameter

��

Accumulation regime parameter �, �

Weights of short- and medium-run inflation( � (1�

� �)��)

� Weight with respect to backward and forwardlooking expectations

�, ��, ��

Tax rate (of capitalists and workers)� const. (ort�� (T� rB/p)/K� const.,t�� (T

�� rB/p)/K� const.)

s�

Savings ratio (out of profits and interest)s�

Savings ratio (out of wages, � 0 in this book)�

Fiscal policy parameter

C Mathematical notationx� Time derivative of a variable xx Growth rate of xl�, l

�Total and partial derivatives

y�� y�(l)l

�Composite derivatives

r�, etc. Steady state values (r� a parameter which may differ

from r�)

l�L/K, etc. Real variables in intensive formm�M/(pK), etc. Nominal variables in intensive form

xxiv Notation

General introduction

In this book we shall be concerned with the foundations of integratedmacromodels of monetary growth dynamics in disequilibrium as they havebeen laid out (to some extent) in the sixties and the seventies. Thesefoundations are reconsidered and reformulated as well as extended into auniform and systematic body of macrodynamic models of closed econo-mies with five markets and three agents. The stress here lies on disequilib-rium models as we believe that there is an urgent need for progress in thisneglected, but nevertheless very relevant, area of macrodynamics. We donot believe that the numerous equilibrium models of monetary growththat have been developed over the last two decades� will realize theirpotential for policy analysis if they are not supplemented and confrontedwith disequilibrium analyses that try to portray, with more and moredescriptive exactness and analytical rigor, the macroeconomy and thepolicy scenarios to be investigated.On the one hand completeness of such models is necessary when one

wants to provide a systematic and comparative study of them (and theirpros and cons) which can then be used as a framework and as a foundationfor the further systematic development of this area of macroeconomics.Such a systematic development is almost nonexistent in the literature ondisequilibrium monetary growth dynamics. Partial models may of coursebe of great interest if, as is generally the case, more specialized questions areconsidered.Yet, it should in principle always be possible to trace backwhattype of model has been specialized in such a study and what the generalmodel may look like.On the other hand completeness of monetary growth models with

respect to agents and sectoral behavioral descriptions including budgetrestrictions and with respect to markets and their type of adjustmentprocess is nowadays a compelling prerequisite for a broader acceptance of

� See the survey by Orphanides and Solow (1990) for example.

1

so-called macro ad-hoc (or descriptive) macromodels, to be distinguishedfrom micro ad-hoc macromodels (where ad-hoc refers to the empiricalrelevance of the micro assumptions that are made). The advantage of suchapproaches to macro theory indeed lies in the fact that these models canmore easily be made complete, and thereby tested with respect to thedegree of consistency that is achieved by them, than themanymicro ad-hocmodels that are now the fashion. These latter models are generally partialin nature because of the restrictions that are caused by the technicalcomplexity of the dynamic intertemporal optimization framework thatthey employ. Furthermore, the dynamics of such models is by technicalnecessity generally limited to a study of linearized systems around steadystates. Such approaches automatically exclude the type of complex behav-ior which the models in this book can display.Complete or integrated macrodynamic models therefore may provide a

macro foundation for micro perspectives and be further developed in thelight of the achievements obtained from suchmicro perspectives. Completedisequilibriummacrodynamicmodels of monetary growth therefore main-ly serve the purpose of providing right from the outset a full picture of theeconomy in states of disequilibrium by means of more or less traditionaltools or modules. These modules may subsequently be updated step bystep as better descriptions of their micro foundations become available.As we shall see in this book, there exists now a hierarchically structured

class of such models which build upon each other in a step-by-step im-provement of the modules they contain. Yet, even though at the end of thisbook we will be higher up in the hierarchy of our models, there will remainsome module formulations that are obviously problematic and whichtherefore call for significant further improvement. There is thus the need toextend much further the project begun here. Yet, it should have becomeobvious to the reader by this stage that such a task can be accomplished bycontinuing to proceed in the manner we have developed in this book. Thiswill indeed give rise to a structured body of theories of monetary growth indisequilibrium, up to the most recent developments of disequilibriummacrodynamics, where insights of earlier achievements are preserved andwhere a pathway of systematic progress to more convincing and realisticmodel types becomes visible. This is themain advantage of amethodwhichprovides a class of monetary disequilibrium growth macromodels that allattempt to be complete and thereby clearly show the path to their furtherimprovement and the next required step to be taken in their furtherdevelopment.The resulting prototypes of such models in this book are descriptive in

the sense that they generally use traditional macro tools to describe thebehavior of the various sectors of their economies. These tools may never-

2 The Dynamics of Keynesian Monetary Growth

theless simplify the considered behavior significantly with respect to itsdescriptive content in order to allow us to proceed from simple buildingblocks to more elaborate ones in a systematic fashion, thereby filling theirdescriptive or ad-hoc macro assumptions with more realism step by step.Descriptive components of such macromodels can therefore at first befairly abstract and stylized in their ‘‘descriptive’’ content, due in particularto the tradition that has been established in the formulation of suchcomponents of macromodels. The basic justification for the use of such(sometimes radically simplified) building blocks is that also in this area ofmacroeconomics one has to start from known model structures and to gofrom the simple (and abstract) to the more complex (and concrete) bymeans of a stepwise improvement in the formulation and the analysis ofintentionally complete models of monetary growth.We shall make no attempt here to base the descriptive components of

our models on micro assumptions surrounding the concept of representa-tive agents as is now the fashion in macroeconomics,� since our central aimis a complete presentation and analysis of the interaction of the threesectors of our economies. This interaction will be made more refined as thebook proceeds, leaving a systematic improvement of the behavior of sec-tors to later studies of these models (where also refinements by means ofmodern microfounded approaches may be taken into consideration).The intention of this book on descriptive macrodynamic models thus is

to start from the traditional roots of a more or less orthodox formulation ofsuch monetary growth dynamics (in particular Tobin and Keynes—Wick-sell models of monetary growth) in order to obtain from them and theirdetailed presentation and discussion (from the beginning of chapter 4 of thebook) a description of a general prototype model which may properly beregarded as a Keynesian one. Such a model, which allows (as should be thecase in a Keynesian model) for the investigation of unemployed labor aswell as underutilized capital, has rarely been considered in the literature,and certainly not in the fully specified dynamic framework which we shallemploy throughout this book.Instead, a so-called neoclassical production function, and the marginal

productivity postulate for the employment of labor, have generally beenincluded in the existing analyses of monetary growth in such a way thatonly labor is considered as experiencing unemployment (due to nominalwage rigidities). In chapter 5 we shall also allow for neoclassical smooth

� Note, however, that most of the (traditional) behavioral relationships we employ havereceived some micro foundations in the course of their use in macroeconomics.

The exception to this is provided by models of the so-called neo-Keynesian or non-Walrasian type which, however, are seldom as complete as our development andpresentation of the working model of Keynesian monetary growth.

3General introduction

factor substitution and then demonstrate that this does not prevent theanalysis of underutilized capital in a Keynesian setup. In general, however,we will stick to the simpler assumption of fixed proportions in production,since this makes the Keynesian analysis of underutilized resources muchmore transparent.After providing some numerical investigations of the considered models

of Tobin, Keynes—Wicksell and Keynes(ian) type with or without smoothfactor substitution we shall finally consider two further important exten-sions of the Keynesian prototype introduced here: a Metzlerian extensionof this prototype when IS-disequilibrium is allowed for and a ‘‘Marxian’’extension of it which avoids the use of ‘‘natural’’ economic magnitudes asmuch as possible. The final chapter will also point to a variety of omissionsin the modeling framework presented here which must be addressed inorder to properly make the analysis a Keynesian one, particularly since thebehavior of wealth owners is still much too passively modeled in theapproaches to monetary growth dynamics presented in this book. Also,investment behavior is still presented far too simply to portray accuratelythe trade cycle vision of Keynes’ General Theory. All of these extensions,however, must be left for future research.We shall consider throughout this book only macroeconomic models

which fit into the standard and basic framework of a closed three-sectoreconomy (households, firms, and government), where there exist five dis-tinct markets (for labor, goods, money, bonds [savings deposits], andequities [perfect substitutes of bonds]).Moneymarket transactions are, ofcourse, a mirror image of transactions on the remaining four markets andare to be related to these activities by means of budget restrictions for thethree sectors assumed. In table I.1 we use the index d to denote ‘‘quantitiesdemanded’’ and no index in the case of ‘‘quantities supplied.’’ Furthermore,since we will use continuous-timemodels throughout this book we have todistinguish between flow and stock demand and supply since we herefollow the macroeconomic tradition which distinguishes between stockand flow constraints in such a setup; see Turnovsky (1977a) and Sargent(1987) for details. This said, the symbols in table I.1 should be clear as to

We restrict ourselves to this standard, basic framework due to our intention to stay, at leastinitially, very close to orthodox foundations of neoclassical and Keynesian dynamics. Thefollowing modeling framework is therefore chosen, initially, as identical to the one that isemployed in such a conventional textbook of macroeconomics as that of Sargent (1987); seealsoTurnovsky (1977a) for a related framework. In this bookwe shall revise only someof theassumptions (but nevertheless very important ones) that underlie the Sargent approach tocomplete, or integrated, macroeconomicmodels. In some respects the contents of this bookmay thus be characterized as providing simply improved and dynamic counterparts of thethreemodel prototypes that are at the core of Sargent’s (1987, part I)mainly static analysis ofthem.


Table I.1. Basic structure of closed economies

Labor Goods Money Bonds Equitiesmarket market market market market

Households L C M,M� B,B� E,E� Firms: L Y, I� �K — — E,E�Government — G M,M� B,B� —

their economic meaning (a detailed list of the notation employed is pro-vided at the front of this book).�Table I.1 shows the basic structure of the closed economies that are

considered throughout this book and it is of the same type as the one inSargent (1987, chs. 1—5) as will become apparent from its further descrip-tion in chapter 1.We will model the behavior of our three economic agents in the usual

fashion by staying close to behavioral assumptions which are firmly rootedin the tradition of descriptive macroeconomics. This guarantees that themodels considered in chapters 2—7 will not depart too much from theestablished formulations of (textbook) macrodynamicmodels, though theywill be generalized considerably with respect to their degree of integration.As in Turnovsky (1977a), our main aim is to develop and analyze suchintegrated (or complete) models of monetary growth (of closed economies)in a systematic way. In this respect it is of particular importance that thebudget restrictions (BR) of all three sectors, households, firms, government(to be denoted by HBR, FBR, and GBR, respectively), are always fullyspecified. The behavior of the agents that is assumed to take place withinthese budget restrictions may, due to the traditional roots of our modelingframework, still not be too convincing. Yet, improved assumptions orderivations for the assumed behavioral relationship can easily be insertedinto the complete models employed in this book, thereby changing thedescription of one or more sectors of the model, but not the overallformulation of the interaction of these sectors. Our conjecture is that suchimprovements may change details in the models’ behavior but not thegeneral finding of this book that the considered models of monetarygrowth do exhibit a high potential for generating undamped and, if appro-priate nonlinearities are assumed, also viable patterns of cyclical growth.

� Planned aggregate demand Y is, as usual, given by C� I� �K�G. Note also that tableI.1 suggests (again as is customary) that money holdings of firms are considered asunimportant and thus ignored and that there is no bond supply on the side of firms, but onlyequity financing if necessary.


We start with the most orthodox model of monetary growth that isavailable in descriptive macroeconomics: the Tobin (1965) extension of theneoclassical growth model which introduces money as a further asset intothis otherwise purely real framework. This model will be introduced inchapter 2 in a form that is convenient both with respect to our generalassumption of fixed proportions in production as well as from the point ofview of the historical development of capitalistic economies.Our particularreformulation of this basic Tobin model will be extended in various direc-tions in chapter 2 leading eventually to a very general formulation of it thatserves as a basis for our subsequent introduction of a general model ofKeynes—Wicksell type (chapter 3) and later of proper Keynesian type(chapter 4).The general Tobin model is, however, problematic in its assumption of

money-market disequilibrium and the price-adjustment equation that isbuilt upon it. Furthermore its view of the behavior of the firm sector isextremely limited, since it allows only for production decisions in thissubstructure of the economy. All these weaknesses are overcome (in chap-ter 3) by our next prototype, the Keynes—Wicksell approach to monetarygrowth dynamics. Here investment decisions of firms and their financingbymeans of equities are considered explicitly and made consistent with theother sectors of the economy. Price adjustment is also put on a firmer basisin this model type and gives rise to the famous growth cycle mechanism ofGoodwin and Rose as part of this extended framework of analysis and itsdynamical implications. The inclusion of these Classical growth cyclemechanisms, by way of an improved wage—price module of the model, in arelatively pure form, is the main contribution that we will obtain from thisvariation of the Tobin monetary growth model.Having improved the presentation of asset markets (in particular by

assuming money market equilibrium throughout), the goods market re-mains a problem, since the added description of the investment behavior offirms generally now gives rise to a disequilibrium situation which is notpresent in the Tobin approach due to its dependence on Say’s Law on themarket for goods. The further development of the model (in chapter 4) to abasically Keynesian one therefore now adds IS-equilibrium. Followingfrom this latter assumption (and the assumed wage/price adjustment be-havior) the degree of utilization of the capital stock becomes the variablewhich will always adjust appropriately in order to make possible theassumed goods market equilibrium. In contrast to the fashionable fullequilibrium version of the Tobin models we thereby arrive at the basicKeynesian prototype structure that will underlie all following generaliz-ations of models of monetary growth exhibiting IS—LM-equilibrium anddisequilibrium on the labor market and within firms. These disequilibria


are then used as the basis for wage and price adjustments and the invest-ment decision of firms.This latter Keynesian prototype will be extended in various directions in

chapters 4—7 to allow for factor substitution, technological change, wagetaxation, p-star expectations, delayed quantity adjustments, endogenousnatural rates and insider—outsider effects in the labor market. By the end ofthese extensions the Keynesian prototypewill have becomewhat we label aworking Keynesian model. It will also be demonstrated to the reader thatthis working model still represents only a starting point (though already afairly elaborate and consistent one) to a thorough consideration of manyfurther extensions. Indeed section 7.7 provides a survey of such, necessary,extensions.The way in which the basic ‘‘proper’’ Keynesian prototype, and then the

working Keynesian model of monetary growth, is established here will inaddition show that this model type overcomes important weaknesses of thepredecessor models of Tobin and Keynes—Wicksell type by a systematicvariation of them. Nevertheless, each of these two predecessor models isalso of importance in its own right, due to the specific topics that have beenconsidered importantwithin these earlier prototypes. TheTobinmodel, forexample, distinguishes between actual and perceived disposable income ofhouseholds and allows consideration of a number of interesting effects thatflow from this distinction, including the fact that it will represent a non-linear model (again due to this distinction) even if all of its structuralequations are linear. The consequences of distinguishing between actualand perceived disposable incomewill only be considered in chapter 2, whilelater chapters will again identify perceived with actual disposable income,leaving this specificity of the Tobin approach for the later investigations ofour other models.In our presentation of the various model types we shall mostly employ

linear economic behavioral relationships. Thus nonlinearities that appearin the dynamic laws will be naturally occurring in that they are broughtabout by product terms such as the wage bill, state variable quotients suchas the rate of employment, and some formulations being in terms of rates ofgrowth. This serves the purpose of investigating the dynamical systemsthat are implied at first only in a ‘‘naturally’’ or ‘‘intrinsic’’ nonlinear setupin order to see how much ‘‘dynamical complexity’’ is already involved onthis most basic level of the study of integrated economic systems. Occa-sionallywe introduce, however, specific nonlinear behavioral relationships,in particular in investment functions and Phillips curves, in order tomaintain economic viability of the dynamics being analyzed. However, weleave for future research a systematic study of the introduction into ourgeneralmodeling framework of these and other nonlinear economic behav-


ioral relationships which have been proposed in the literature on macro-economic fluctuations.Throughout this book we model expectations as a weighted sum of

‘‘backward looking’’ and ‘‘forward looking’’ components. We endow ouragents with neither the information of the model structure in which theyplay out their economic roles, nor the computational ability that theywould need to form expectations in a way that is currently referred to as‘‘rational’’ in a large body of literature. Our reasons for adopting thisapproach are detailed in section 1.6. In essence these reasons revolvearound a critique of the so-called jump-variable technique which theadoption of a ‘‘rational’’ expectations approach would necessitate as wellas a growing body of empirical evidence which suggests that our approachto expectations modeling may be more appropriate. However here westress that the future research agenda to which we have already referredwill need to incorporate the effects of heterogeneity of expectations and oflearning on the part of the various economic agents of our models.This concludes the description of the basic line of reasoning that we will

employ in the development of our model structure. Since there is a clearprogression frommodel to model in this book we will generally explain themodel equations only when they appear for the first time. Before we nowproceed to such a systematic step-by-step development of prototypemodels of monetary growth we will briefly consider in chapter 1 certainroots of these approaches in the literature.The material presented in chapters 1 to 7 of this book is neither of direct

textbook type nor written in the way of a handbook on monetary growth.There is now a variety of advanced textbooks on macroeconomics avail-able, ranging from traditional Keynesian analysis of extended IS—LM typeto analysis that claim to go ‘‘beyond IS—LM,’’ see in particular Blanchardand Fischer (1989), Carlin and Soskice (1990), Karakitsos (1992), Leslie(1993), Turnovsky (1995), and Romer (1996). On the one hand, we add tothese presentations a new hierarchically structured set of theories andmodels of monetary growth that can be used for classroom teaching. Onthe other hand, we seek to draw to the attention of writers of advancedtextbooks and researchers in the field of macrodynamics the fact thattraditional analyses of models of monetary growth cannot be viewed as aset of isolated models. Rather they must be considered as a lively body ofsystematic studies which, when fully integrated, are still poorly understoodand where further investigation will provide a firm foundation and a betterunderstanding of existing and future developments in this area.This book is also not a handbook on aspects of monetary economics or

more precisely a survey on the theory of monetary growth, as provided, forexample, by Orphanides and Solow (1990). Instead we show that there still


exists a large evolutionary potential in traditional macrodynamics thatleads us to integrated macrodynamical models with Keynesian short-runfeatures and Keynesian and monetarist features in the medium run as wellas in the long run. These integrated models not only allow us to evaluatethe contributions of the two schools of economic thought from this integ-rated perspective, but also serve to put into perspective more recent contri-butions to the theory of fluctuations and growth in monetary economies.In this way our book provides a benchmark against which alternativeapproaches can be judged and be developed further, including the workingmodel of this book, towards a common core of macrodynamics that ‘‘we allcan believe in.’’


1 Traditional monetary growthdynamics

1.1 Introduction

We reconsider in this chapter the leftover ruins of traditional monetarygrowth dynamics� which, with respect to the general dynamics they cangive rise to, have so far been poorly analyzed and understood in theliterature.�We attempt to show to the reader, in section 1.2 in overview, and in

detail in chapters 2—5, that these leftover ruins can be arranged andrepresented in a systematic way so that they form a hierarchical structuredclass of monetary growth models where each subsequent model typeeliminates some of the weaknesses of the preceding model type. We thenindicate in section 1.2 two ways in which this methodological approach tomacrodynamics can be significantly extended beyond the existing scope oftraditional models of monetary growth. Firstly, this way of proceeding infact leads to the establishment of a proper (still traditional), but muchneglected Keynesian model of monetary growth where both labor andcapital exhibit fluctuating degrees of utilization independently of the as-sumptions that are made on ‘‘technology.’’ Secondly, our approach leads toa further improvement of this IS—LMgrowth type of dynamics by allowingfor sluggish price, as well as quantity, adjustments (two Phillips-curvemechanisms and a Metzlerian treatment of disappointed sales expecta-tions) and by establishing thereby what we will call the working Keynesianmodel of this book. Section 1.2 therefore provides a survey of what we callthe macro foundations of (disequilibrium) macroeconomics, namely theindication that there is a systematic way of proceeding from less sound and

� See Turnovsky (1995, part I) with respect to another reconsideration of integratedmacrodynamics of traditional type.

� See for example Sargent’s (1987, ch. 5) analysis of ‘‘Keynesian Dynamics’’ of AS—AD typeand its reconsideration in Flaschel (1993, ch. 6—7) and Franke (1992a), or Stein’s (1982)investigation of dynamic models of Keynes—Wicksell type and its reconsideration inFlaschel, Franke, and Semmler (1997, ch. 10).

10

elaborate to more sound and elaborate presentations of integrated macro-dynamic models of monetary growth. In this way we demonstrate both inthis survey chapter and in more detail in chapters 2—7 a systematic pro-cedure on the macro level by which integrated or complete macrodynami-cal models can be made more and more elaborate and coherent in theirpresentation of the fundamental feedback structures that characterize in-terdependence on the macro level.Our approach proceeds independently of any justified claim for better

micro foundations of macroeconomics. Indeed, improved micro founda-tions that emerge from research in this area should be capable of integra-tion into appropriate modules of the macro structure that we build in thisbook. There is no space here, however, to go into this topic in detail. Arecent approach which considers the problem of the micro foundations ofmacroeconomics from a critical perspective and which provides alternativeand interesting micro foundations of macroeconomics (not based only onbudget and technological constraints) is Hahn and Solow (1995). There it isfound in particular that stickiness of wages and prices may be good foreconomic stability. Our treatment of Keynesian monetary growth in chap-ters 4—6 arrives at a similar conclusion, but from a quite different perspec-tive.The various steps in the building of a class of hierarchical structured

models of monetary growth are made on the basis of assumptions on thestructure of markets and sectors of the economy as they are used in Sargent(1987, chs. 1—5) which are indeed very convenient for the first stage of theproject started in this book.We extend Sargent’s (1987, part I) mainly staticanalysis of AS—AD macroeconomics (cum growth in his ch. 5) on the onehand into a full dynamic analysis of growing monetary economies and onthe other hand into the direction of proper Keynesian models of monetarygrowth (where also firms are no longer on their supply schedule). In thisway we lay foundations for a Keynesian approach to monetary growthwhich has rarely been studied in the literature so far.When the final stageis reached in this book, however, the need for further extensions in thestructure of Keynesian monetary growth dynamics will become apparent.Possibilities for such extensions are briefly discussed at the end of chapter7. These provide a research agenda of systematic developments along themethodological lines established in this book.In sections 3—5 of this chapter we consider for introductory purposes

basic models of Tobin, Keynes—Wicksell and AS—AD type. Very generalversions of these approaches are introduced and investigated in subsequent

See for example Orphanides and Solow (1990), where models of this type are not evenmentioned.

See Chiarella and Flaschel (1998f ) and Chiarella et al. (1998, 1999).

11Traditional monetary growth dynamics

chapters. Since Sargent’s treatment of the AS—ADgrowthmodel focuses onthe role various expectations schemes play in the dynamics generated bythismodel type, we in addition provide an alternative view on themodelingof expectations in section 1.6 which will be used in subsequent chapters onvarious levels of generality. Section 7 provides a few characterizationsconcerning the proper Keynesianmodels of monetary growth that we shallintroduce and analyze in chapters 4—7.

1.2 Macro foundations of macroeconomics

The purpose of this section is to indicate to the reader that there is ahierarchical structured body of disequilibriummodels of monetary growthwhere each subsequent stage in the development of such models improvesthe descriptive relevance of the preceding stage in a systematic and signifi-cant way. Independently of the need for sound micro foundations of theassumed (fairly conventional) behavioral relationships, the evolution ofdisequilibrium macrodynamics (which is not easily micro founded) doesthereby indeed exhibit systematic progress to more and more convincingformulations of the fundamental modules of the dynamics of monetarygrowth in disequilibrium, and thus to the description and analysis of realgrowth dynamics. To show this in detail and to indicate how the frame-work of disequilibrium macrodynamics that we develop can be extendedbeyond its current scope are two of the main purposes of this book.Subsection 1.2.1 provides a brief summary of the evolution and achieve-

ments of dynamic disequilibrium models of monetary growth in the past.Subsection 1.2.2 reviews the contributions that this book will make to thecurrent state of the theory of monetary growth with under- or overem-ployed factors of production. A brief outlook on what needs to be and canbe done on the basis of the results achieved in this book will conclude thesubsection.

1.2.1 A brief genesis of disequilibrium models of monetary growth

We discuss in this subsection forerunners to the Keynesian model ofmonetary growth to be introduced in chapter 4 and developed further insubsequent chapters toward our working model of disequilibrium monet-ary growth dynamics.

The starting point• Neoclassical models of monetary growth of Tobin type• Extension of the Solow model of real growth:

— towards an inclusion of financial assets


— where money market (dis-)equilibrium drives inflation— in interaction with inflationary expectations

Neoclassical models of monetary growth were introduced into the macro-economic literature through the work of Tobin (1955, 1965) which ex-tended the Solow model of real growth by introducing monetary factors.Generalized versions of this model type were developed subsequently byJohnson (1966), Sidrauski (1967a), Hadjimichalakis (1971a,b), Nagatani(1970), Hadjimichalakis and Okuguchi (1979), and Hayakawa (1979). Bur-meister and Dobell (1970), Sijben (1977), Sargent (1987), and Orphanidesand Solow (1990) give further presentations and a survey of this literature.These extensions were generally characterized by the consideration ofmoney as an asset in addition to real capital and the use of money marketdisequilibrium as the foundation of the theory of inflation and inflationaryexpectations, coupled with Say’s Law on the markets for goods (therebyexcluding any goods-market problems).�Tobin type models have led to anenormous amount of literature on equilibrium growth models with opti-mizing behavior of economic agents, which, due to its general equilibriumnature, is not a suitable topic for a book such as this, the focus of which ison disequilibrium monetary growth theory.Typical issues addressed by the above-cited authors were the analysis of

the steady state effects of the growth rate of the money supply (and ofso-called Tobin effects) and the local stability analysis of the steady statewhere in particular the destabilizing role of inflationary expectations wasinvestigated when the adjustment of adaptively formed inflationary expec-tations became sufficiently fast. A detailed presentation of such stabilityissues is provided in Hayakawa (1984), while Benhabib and Miyao (1981)investigate the possibility of the cycles generated by Hopf bifurcations forintermediate adjustment speeds of inflationary expectations. It is shownthereby that the Cagan (1967) inflationary dynamics and the disequilib-rium approaches that were built on it by Goldman (1972) and others notonly give rise locally to saddlepath situations (that are now the basis of thejump-variable technique of rational expectations models), but that therewill emerge limit cycles for particular ranges in the adjustment speed ofexpectations from the nonlinear structure of these neoclassical models ofmonetary growth.In chapter 2 we will start from the most basic (general equilibrium)

version of the Tobin monetary growth model and shall subsequently

� Labor market phenomena were generally treated as in Solow (1956) by assuming fullemployment and the macroeconomic marginal productivity theory of income distribution.But labor market disequilibrium is easily introduced into this framework as in Goodwin(1967), here combined with neoclassical smooth factor substitution as discussed in detail inchapter 5.


establish step by step a general disequilibrium version of this model type(with money and bonds as financial assets and, of course, Say’s Lawremaining the [trivial] representation of goods market equilibrium). As inthe evolution of the literature on models of monetary growth, we thusbegin this book with the stability problems of monetarymodels of neoclas-sical growth where we, however, attempt to stress the cyclical properties ofthe dynamics of these models which, when necessary, may be bounded andthus imply viable cyclical oscillations through appropriate nonlinearitiesin the assumed behavioral relationships of neoclassical monetary growth.The main ingredients of our development of a Tobin general disequilib-rium monetary growth model are listed at the head of this subsection.

The Keynes—Wicksell alternative• Independent investment behavior based on Tobin’s q• Removal of Say’s Law due to savings� investment• Augmented Wicksellian demand-pressure price-inflation• Money wage Phillips curve• Full capacity growth

The next model type that we develop is based on the Keynes—Wicksellapproach to monetary growth. The most important work in this area ofmonetary growth theory has been provided in the late sixties and earlyseventies by Stein (1966, 1968, 1969, 1970, 1971) and by Rose (1966, 1967,1969). Further contributions are Fischer (1972), Fujino (1974), Sijben(1977), Nagatani (1978), Brems (1980), Iwai (1981), and Asada (1991). Inparticular Stein (1982) has related this type of approach to the discussionbetween Keynesians, monetarists, and New Classicals, while Skott(1989a,b) provides a general theory of conflict about income distributionand of effective demand with similarities to the Keynes—Wicksell theory ofmonetary growth. Rose (1990) pursues the same aim from a somewhatdifferent perspective and relates his general approach to many partialmodels of macroeconomic dynamics. The work of these latter two authorsshows that there are still emerging important developments of this Keynes-oriented area of monetary growth theory, which in particular attempts toprovide a Marshallian perspective of Keynes’ theory of effective demand.The importance of the Keynes—Wicksell approach toward an explana-

tion of the working of a (growing) monetary economy stems from twoobservations. Firstly, in the recognition that savings and investment deci-sions are to be differentiated from each other in an essential way (therebydenying the validity of Say’s Law both in its trivial and in a more elaborateform), and that the theory of price inflation must be related to the goodsmarket and its disequilibrium and not, as in the generalized Tobin models,


to money market disequilibrium. Goods market imbalance was measuredin these approaches through the deviation of investment decisions fromsavings decisions. The theory of inflation was based on this imbalance andaugmented by expected inflation in Fischer (1972) in a monetarist fashionin order to allow for steady state equilibrium. The money market, bycontrast, was now described through the usual Keynesian LM-equilibriumcondition as the theory of the nominal rate of interest (the deviation ofwhich from the nominal gross rate of profit was then used to determine thelevel of investment).Secondly, that this alternative to the neoclassical view on monetary

growth dynamics stressed the cyclical implications of labor market dis-equilibrium, the conflict over income distribution and capital accumula-tion. The work of Rose (1967) in particular established the Goodwinian(1967) growth cycle mechanism in an independent way and from a differ-ent perspective by relating it to a locally unstable Wicksellian theory ofprice inflation that gave rise to persistent fluctuations by way of appropri-ate assumptions on wage flexibility in a setup with smooth factor substitu-tion. Rose (1990) provides important extensions of this type of monetarygrowth theory, extensions which have significantly influenced the formu-lation of the wage—price dynamics of our general Keynes—Wicksell modelin chapter 3.Our view on the Keynes—Wicksell contribution to the analysis of monet-

arygrowth is that it representsadecisive step forward in themacroeconomicdescription of such growth processes. Neoclassical and Keynes—Wicksellmodels of monetary growth are not situated in the hierarchy of monetarygrowthmodels on the same level of abstraction, but follow each other in thisorder, since the latter model type takes account of the independence ofinvestment decisions from savings conditions and tries to incorporate thisfact from a Wicksellian perspective in the simplest way possible. Thisextension in the approach tomonetarygrowthalso leads to an inclusionof anew financial asset besides money and bonds (equities) that is explicitlyintroduced, and related to Tobin’s q, in our general reformulation of theKeynes—Wicksell approach in chapter 3. The main ingredients in ourdevelopment of the Keynes—Wicksell alternative are listed on p. 14.The Keynes—Wicksell type of analysis exhibits a number of problems.

Firstly, the labor market is treated basically from a (neo)classical perspec-tive in the same fashion as in the Goodwin model of the classical growthcycle, except with neoclassical factor substitution. Secondly, goods marketimbalances drive prices and not quantities as in the Keynesian dynamicmultiplier approach. Thirdly, capital is always operated at its full capacityas described by the usual profit-maximizing marginal productivity condi-tion for real wages. Therefore, the (neo)classical view on capital accumula-


tion is still, at least partially, present in this type of analysis. A logicallycompelling next step in the Keynesian analysis of monetary growth, there-fore, should be to establish a proper type of Keynesian goods market andmoney market analysis in the tradition of the IS—LM approach to thedescription of the functioning of these two markets. This is indeed the stepthat was taken in the literature on Keynesian dynamics in the context ofmonetary growth in the late seventies.

The Keynesian AS—AD growth model• Keynesian IS—equilibrium (in addition to LM—equilibrium)• Infinitely flexible prices based on marginal wage costs• Expectations — augmented money wage Phillips curve• Profit-maximizing output decisions of firms

The textbook treatment of Keynesian monetary growth dynamics (seeTurnovsky 1977a and Sargent 1987 for typical examples) dispensed withthe Wicksellian approach to the determination of the price level, or ratherits rate of change, by simply adding wage dynamics and inflationaryexpectations dynamics in a monetarist fashion and Solovian capital stockgrowth to the usual AS—AD approach of the Keynesian short-run macro-economic equilibrium.� To date this AS—AD growth model has beenconsidered as the representation of traditional Keynesian growth dynami-cs (see for example Turnovsky 1995, part I). Yet the fact remains that thefull dynamics of such integrated AS—AD growth models have rarely beenanalyzed to a satisfactory degree, which means that the dynamic behaviorof these seemingly conventional models is poorly understood. This repre-sents an important gap in the theory of monetary growth, since we there-fore do not have a generally accepted pool of knowledge at our disposalagainst which the achievements of more recent theories of monetarygrowth can be usefully compared. The main elements of the KeynesianAS—AD growth model are summarized above.There are, however, inconsistencies present in the AS—AD theory of

effective demand, inflationary dynamics, and real capital accumulation.Basing the theory of the price level on its determination through marginalwage costs, as Keynes (1936) did, amounts to assuming that producers are,on the one hand, constrained by the effective demand for goods, but are, onthe other hand, capable of passing on this constraint to the labor supply ofhouseholds, by allowing in one way or another for profit-maximizingprices so that they can stay on their supply schedule. In our view this

� IS—LM equilibrium coupled with the assumption that prices are always equal to marginalwage costs, the money wage level being given at each point in time.

See in particular Flaschel, Franke, and Semmler (1997) for some investigations of thedynamics of these AS—AD growth models.


basically means that firms are price takers and quantity takers at one andthe same time, which would give rise to a contradiction if prices are notassumed to adjust in such a way that the level of effective demand becomesconsistent with the profit maximizing level of the output of firms. Ourconclusion is that the Keynesian theory of goods-market constrained firmsneeds a theory of the price level other than that of the neoclassical ap-proach to the theory of the firm. Such an alternative theorymight be that ofmonopolistic competition or even more advanced theories representingmore advanced stages in the evolution of capitalistic market economies.Furthermore, a Keynesian theory of the AS—AD type (even if it wereconsistent) would still be a theory of full-capacity growth and would thusrepresent only a partial description of what we observe in reality.Barro (1994b, p. 4) has recently come to the same conclusion from a

different but related perspective, stating in particular:

We have available, at this time, two types of internally consistent models that allowfor cyclical interactions between monetary and real variables. The conventionalIS—LM model achieves this interaction by assuming that the price level and thenominal wage rate are typically too high and adjust only gradually toward theirmarket-clearing values. The market-clearing models with incomplete informationget this interaction by assuming that people have imperfect knowledge about thegeneral price level.

This quotation lends further weight to our viewpoint that models ofIS—LM growth with gradually adjusting wages and prices are the correctalternative to the general equilibrium approach to monetary growth. Thisperspective is in fact not a new one, but has indeed been essential for theso-called neo-Keynesian or non-Walrasian disequilibrium analysis of theshort-, the medium-, and sometimes also the long-run evolution of tempor-ary fixed price equilibria. This approach can therefore be used to improveconsiderably the presentation of macrodynamic disequilibrium growth ofAS—AD type, though most of the efforts in this area have gone into themodeling of fixed price temporary equilibria from a microeconomic pointof view (which are not reviewed in the following characterization of thisapproach).

Neo-Keynesian monetary growth analysis• Three regimes of the IS—LM model: the Keynesian regime,the Classical regime, and repressed inflation

• No full capacity growth in the first and the last regime• Varying capital utilization rates and price dynamics• Varying labor utilization rates and wage dynamics• Sluggish wage- as well as price-level adjustments


• Extended investment functions based on profitabilitymeasures and the rate of capacity utilization

From the macroeconomic point of view, the work of Benassy and Malin-vaud is here of special interest for the purposes of this book, see inparticular Benassy (1986b) and Malinvaud (1980), and from the viewpointof monetary growth theory also the collection of essays in Henin andMichel (1982). Benassy (1986b) provides in particular a detailed present-tion of the three regimes that may be of particular relevance in the analysisof macroeconomic temporary equilibrium positions and dynamic modelsof inflation, the business cycle and the role of expectations based on thisthree-regime analysis. Malinvaud (1980) considers investment behaviorbased on profitability and capacity utilization besides a consumptionfunction that is typical for the fixed price approach to temporary equilib-rium. He incorporates these behavioral relationships into a three-regimemedium-run model of Keynesian depressions, under the additional as-sumption that the profitability effect of real wage changes is less significantthan the consumption effect of such changes.We will borrow from this literature two important ideas. The first one,

which also appears in the quite different macrodynamic approach of Rose(1990), is that there should be two Phillips-type curves in a Keynesianmacrodynamic model, one for the wage level and one for the price level.Both of these are expectations augmented (from a cost-push perspective),and both exhibit demand pressure components that (in the first instance)are to be represented through the utilization rates of the two factors ofproduction, labor and capital. The second idea is that the investmentbehavior of Keynes—Wicksell and dynamic AS—AD models, which wasbased on Tobin’s q solely, should in addition be augmented by capacityconsiderations in order to take account not only of profitability differen-tials, but also of the now varying utilization rate of the capital stock.These are the main elements (summarized in the list above) of the

neo-Keynesian analysis of (the evolution of) fixed price equilibria that wewill use in our formulation of a proper Keynesian model of monetarygrowth with IS-equilibrium or IS-disequilibrium. In this way we overcomeimportant limitations of the growth models of Keynes—Wicksell and theAS—AD growth type by allowing also for fluctuating utilization rates ofcapital as in neo-Keynesian analyses of the medium or the long run.However, we do not make use of the regime-switching methodology ofneo-Keynesian analyses, since we believe that there are significant buffersin the process of capital accumulation that generally prevent the occur-rence of hard kinks caused by either labor demand, in the Classical regime,or labor supply, in the regime of repressed inflation.


1.2.2 Keynesian monetary growth dynamics: new steps in thehierarchical evolution of integrated macrodynamics

In this subsection we discuss the two basic stages in our development of aproper Keynesian model of monetary growth, i.e., the basic prototypemodel of this kind investigated in chapter 4 and the working model wedevelop from it in chapter 6.

Keynesian monetary growth analysis: the basic prototype• Only the Keynesian regime of IS—LM growth• Based on excess capacities for labor and capital• NAIRU-type rate of employment and of capacity utilization• No normal capacity growth of output outside the steady state• Varying capital utilization rates and sluggish price leveldynamics

• Varying labor utilization rates and sluggish wage leveldynamics

• Extended investment function based on Tobin’s q and therate of capacity utilization of firms

The new features listed above, taken from neo-Keynesian macrodynamicsthat we have considered on the basis of the ‘‘Neo-Keynesian’’ List on page17, are employed in chapter 4 (and chapter 5) to formulate, on the basis ofthe IS—LM part of AS—AD models, our basic prototype model of Key-nesian monetary growth both with and without smooth factor substitu-tion. As a Keynesian model this should, and now indeed does, exhibitimbalances in the employment of labor as well as capital, independently ofwhether there are fixed proportions in production (chapter 4) or neoclassi-cal smooth factor substitution (as in chapter 5). This new framework forour models of monetary growth overcomes the basic problems of theKeynes—Wicksell model as well as the AS—AD growth model, namely, tomeasure goods-market disequilibrium either by IS-disequilibrium coupledwith full capacity growth, or to have IS-equilibrium at each point in timeand to put the burden of insufficient effective goods demand on the labormarket solely, thereby allowing for full capacity growth as in the Keynes—Wicksell approach (of which it is in fact a limit case for price adjustmentspeeds going to infinity).In our framework, however, labor and capital experience varying utiliz-

ation rates caused by IS—LM equilibrium in the market for goods andmoneywhich drive prices and determine the rate of inflation on the marketfor labor and goods in the fashion of a wage—price spiral (augmentedthrough expectations on wage and price inflation), and which in the case ofcapital also influence the investment behavior of firms. We thus arrive at


the first truly Keynesian model of monetary growth in this book, and onthis level of generality also in the literature. This model is introduced inchapter 4 as a systematic extension and modification of the one-sidedKeynes—Wicksell representation of growth in a monetary economy devel-oped in chapter 3. In view of the earlier quotation from Barro (p. 17) wehave thus arrived at a model type which may be considered as the inter-nally consistent fixed-price IS—LM alternative to the market-clearingmodels of monetary growth which are in vogue today. We note in passingthat this model type is, however, not the final step in the hierarchy ofdisequilibriummodels of monetary growth that we develop and investigatein this book.In describing the new (and also old) building blocks of our Keynesian

disequilibrium model of monetary growth we have only referred to theKeynesian IS—LM regime of the three-regime scenario generally found inmacroeconomic analyses of the neo-Keynesian variety.We have thus madeno reference to the other two regimes of Classical unemployment andrepressed inflation. In contrast to the views in this strand of neo-Keynesianliterature we believe that such an approach is justified from the descriptivepoint of view as well as from the viewpoint of the monetarist reformula-tions of the AS—AD growth model, as presented for example in Sargent(1987, part I). In this respect our model of IS—LM growth with sluggishwages and prices has borrowed from the theory of AS—AD growth withperfectly flexible prices� in that it incorporates not only a ‘‘natural’’ level ofthe employment of labor into its money wage Phillips-curve,� but alsoproceeds in a similar way with respect to the price level Phillips curvepresent in it. This latter Phillips curve is, in chapter 4, no longer based onIS-disequilibrium (which is nonexistent there), but refers to deviations ofthe rate of capacity utilization from a normal or desired rate of capacityutilization (less than one).At and sufficiently near to the steady state of our IS—LM growth model,

therefore, only the Keynesian regime prevails. Situations of repressedinflation or Classical capital shortage may or may not come about far offthe steady state depending on how the (nonlinear) dynamics of this IS—LMgrowth model are formulated far off the steady state of the model. We thusbelieve that it is not sensible to formulate a Keynesian disequilibriummodel of monetary growth from the descriptive point of view in such a waythat, when in steady state, it may, by the slightest conceivable disturbance,just as easily switch into a situation of capital shortage or repressedinflation as into a Keynesian effective demand regime. Capitalist econo-mies most of the time exhibit certain excess capacities on the market for

� Based on marginal wage costs.� In the form of the employment rate complement of the so-called NAIRU.


labor as well as within firms which allow for situations of the overemploy-ment of both factors should there be sufficient effective demand on themarket for goods. A very basic example for this observation, probably notintended to support our claim, is indeed provided in Benassy (1986b, ch.11), where only the IS—LM regime is needed in the derivation of a medium-run wage dynamics (of limit cycle type) within the scope of this regime.There are not many monetary growth models of the neo-Keynesian

variety in the literature; indeed most of them are collected in the citedvolume of Henin andMichel (1982). The reason for this, in our view, is thatsuch growth models are very difficult to formulate and to analyze, due tothe various situations of rationing and the wealth of rationing schemes thatare possible in their dynamics.We shall not here or in the remainder of thisbook go into a discussion of these dynamic models with temporarily fixedprices and wages which treat shortages by hard restrictions (strict inequali-ties) in the place of smooth adjustments in the neighborhood of suchshortages (based on appropriate nonlinear adjustment behavior). We haveargued above that there is no need to proceed along these lines. This iscertainly true from a local perspective, but, in our view, also applicable tomore global types of analysis, once the typical reaction patterns of capital-istic market economies that come about in situations that approach laboror capital shortage are taken into account. Neo-Keynesian models ofmonetary growth have neglected such smooth regime switching processesin the neighborhood of absolute full employment ceilings and have there-fore analyzed situations which are not typical for the process of capitalaccumulation. This is the reason why we do not consider the neo-Keynesian type of monetary growth model in chapter 4 and thereafter.Instead, chapters 4 and 5 will provide various extensions of our proto-

type model of Keynesian monetary growth of chapter 4. These extensions(see page 19 for a summary list) enrich the descriptive relevance of thisprototypemodel, but they do not add much in the direction of a significantfurther level in the hierarchy of disequilibriummodels of monetary growth.It is in chapter 6 that the next decisive step in the systematic evolution ofsuch disequilibrium growth models is taken, in particular motivated bysome strange instability scenarios of the ultra-short-run features of ourIS—LM growth prototype observed in chapter 4 as well as by the im-plausible asymmetric treatment of prices and quantities of that chapter. Asstated above, wages and prices adjust sluggishly in the IS—LM growthmodel of chapter 4, yet quantities, due to the assumed IS—LM equilibrium,adjust with infinite speed to always ensure goods- (and money-) marketequilibrium. Furthermore it is observed in chapter 4 that this equilibriummay be unstable when viewed from a dynamic multiplier perspective. Amore symmetric treatment of the adjustment speeds of prices as well as


quantities may here be preferable, and may lead to more convincing resultsin the explanation of growth and fluctuations. In strict contrast to themarket-clearing approach to monetary growth we thus assume here thatprice, wages, and the output of firms all adjust with finite speed in view ofthe imbalances that are relevant for them. Incorporating this into theapproach of chapter 4 leads to the next stage in the evolution of ourdisequilibrium models of monetary growth.

Keynes—Metzler monetary growth theory• Sluggish wage and price adjustments as in IS—LM growth• Extending IS—LM growth towards IS-disequilibrium• Inventory adjustment mechanism ofMetzlerian type coupledwith a sales expectations mechanism

• The working Keynesian model of monetary growth whichputs the modules of traditional Keynesian macrodynamicstogether

• An internally consistent Keynesian model of monetarygrowth to be comparedwith more recent developments in theliterature

With this stage in the hierarchy of Keynesian models of monetary growthwe arrive at our working Keynesian model, whose main elements areoutlined above. This model overcomes the weaknesses of the still one-sidedIS—LMgrowth dynamics and allows for a variety of results for a Keynesiantheory of fluctuations and growth, as we shall see in chapter 6, from partialperspectives as well as an integrated or total one. Assuming a Metzlerianinventory adjustment process in the place of an instantaneous clearing ofthe market for goods increases the dynamics by two dimensions and leadsto further buffers that can prevent a switch from the Keynesian regime toone of capital or labor shortage. As chapter 6 will show, the stabilityanalysis of the resulting model type is already fairly demanding so thatmany more interesting results on this (from an economic perspective stillbasic) integrated prototype model of monetary growth can be expected tobe obtained from future research.This highest level in the hierarchy of our models of monetary growth is

therefore but the beginning of a meaningful analysis of the Keynesianapproach to the theory of fluctuations and growth. The theory obtainedfrom our integrated perspective is thus still in its infancy. In fact, one mightclaim at the end of this book that our prototype model of Keynes—Metzlertype is but the completion of the oversimplistic Keynes—Wicksell project ofdescribing the macroeconomics of Keynesian monetary growth, since wehave arrived again at IS-disequilibrium as a theory of quantity adjust-


ments, but now without full capacity growth, the implied varying rate ofcapacity utilization being in turn one of the reasons for a changing rate ofinflation.In our view there is an urgent need for a better understanding of the

dynamics of this integrated Keynesian prototype model in order to put atthe disposal of economists a consistent theory of fluctuations, inflation, andgrowth against which the (mostly partial) achievements of newer and moremodern approaches of new- or post-Keynesian type can really be judged.Otherwise, the statement that traditional Keynesian approaches to moneyand growth (which, as we hope to have shown, have not existed so far froma truly integrated perspective) are outdated in their potential to describethe evolution of capitalistic economies, will remain fairly superficial.We will not pursue in this book the task of overhauling the modules of

our working model of chapter 6 from the perspective of the new- orpost-Keynesian theory, proceeding thereby to probably still more ad-vanced models of Keynesian monetary growth, but leave this topic forfuture investigations.�� Insteadwe will provide in the concluding chapter 7,on the basis of a suitable simplification of the Metzlerian inventory mech-anism, a modification of our working model which proposes simple en-dogenous determinations of the NAIRU rate of employment and of thenatural rate of growth of the economy. This modification in additionimproves the formulation of the money wage Phillips curve by takingaccount of insider—outsider effects on the labor market and the possibilityof overtime and short time working within the firm. In this way we provideone final example of how to proceed further in the development of ahierarchy of disequilibrium approaches to monetary growth which bringthe modeling framework closer and closer to a situation where macro-econometric applications become reasonable and also compelling.

Looking ahead• Further module variations of the Keynes—Metzler workingmodel

• Toward the new-Keynesians (monopolistic wage and pricesectors)

• Towards the post-Keynesians (more sophisticated asset mar-kets) or whatever else

• Further module variations: towards modern macroecono-metric model building

By the end of chapter 7 the stage is set for the development of a body ofmore and more consistent and convincing descriptive Keynesian macro-

�� See Chiarella et al. (1999) for work in this direction.


models of monetary growth, on the basis of which a systematic and newanalysis of this type of macrodynamic modeling is within reach. The waythat we have chosen to arrive at this stage represents, in our terminology,the macro foundations of (disequilibrium) macroeconomics, which stressesand develops sophisticated integratedmacrodynamic interaction and feed-back structures in the place of more or less partial microeconomic under-pinnings. Of course, our approach does not deny the value of such under-pinnings. Indeed, research on these needs to proceed in parallel with thedevelopment of the framework we have outlined in order that the variousmacrodynamic interactions and feedbacks can have a firm theoreticalbasis.We therefore start in this book from known (already fairly integrated)

macrodynamic presentations of monetary growth in disequilibrium. Wethen make these models complete models on their respective level ofabstraction and arrange them in a systematic order such that each subse-quent model type can be understood as an improvement of certain import-ant weaknesses of the precedingmodel.We show finally that one can arrivethereby at a model type that allows for laboras well as goods-marketdisequilibrium, both with components that lie inside as well as outside thefirm. We believe that this approach for the first time provides a properstarting point and integrated working model for the Keynesian analysis ofbusiness fluctuations in the utilization rates of both labor and capital andof growth in a monetary framework. We also stress once again that thissynthesis of earlier attempts to provide a truly Keynesianmodel of disequi-librium monetary growth is at the same time a very traditional one. In ourview it is remarkable that this task has been, and is still, very muchneglected in the literature on Keynesian macrodynamics, even though it isnowmore than sixty years since the appearance of Keynes’GeneralTheory.

1.3 Basic Tobin models of monetary growth

This and the following two sections consider in some detail traditionalmacrodynamicmodel building of the Tobin, the Keynes—Wicksell, and theAS—ADgrowth type in order to lay foundations for the generalizedmodelsof this type considered in chapters 2 and 3. Further presentations oftraditional models of monetary growth can be found in Turnovsky(1977a,b, 1995), Sargent (1987), Sijben (1977).Tobin (1955, 1961, 1965) was the first to attempt, and to succeed, in

integrating possible influences of the growth in money supply into the realmodel of growth and capital accumulation of Solow (1956). In particular,his 1965 contribution can be viewed as laying the foundations and posingthe basic questions for models of monetary growth, there in a neoclassical


framework where Say’s Law is assumed to hold (no independent invest-ment function) in a refined form. He added to this neoclassical model aportfolio choice mechanism�� and thereby connected money growth andcapital formation. Other early treatments of monetary growth along thesame lines were provided by Johnson (1966, 1967a,b).This first prototype model of monetary growth is usually presented in

the literature in the following way.��

K� � (M� /p)� s(Y� (M� /p)), (s� const.) (1.1)

M� pYh(��), (h�� 0) (1.2)

Y�F(K,L),� �F�,� �F

�(1.3)

L�L,w��p,L� nL, (n� const.) (1.4)

M� � p(TR�T )�M, (� const.) (1.5)

�� p. (1.6)

In our following brief reconsideration of this model we assume that thereader is familiar with the Solow—Swan model of real economic growthwhich is built on equations (1.1), (1.3), and (1.4) of the above model, butwith the simplified savings function K� � sY in the place of (1.1). This realmodel of economic growth gives rise to the following so-called fundamen-tal equation of economic growth�

k� � sf (k)� nk, k�K/L,

which states that the time-rate of change of the capital intensity, k� , must beequal to the difference between savings per head and nk, the amount ofinvestment needed for pure capital-widening.In this real growthmodel we have no government and no depreciation of

capital. Disposable income of households Y� is therefore simply given byoutput Y. In the Tobin monetary growth model, disposable income ofhouseholds is not so simply handled. Instead of Y��Y, equation (1.1) isused, and is usually explained by stating that real private savings are a fixedproportion of real disposable income and are spent on either capital or realbalances formation:K� or (M� /p) (see Orphanides and Solow 1990, p. 230, forexample). Such a statement is, however, a rather condensed one; what isreally involved in household savings decisions can be made more transpar-ent in the following way.

�� And a particular concept of disposable income, as we shall see below.�� SeeOrphanides and Solow (1990) for details and notation and also pp. xxii—xxiv for a list of

symbols (which are fairly standard in the presently considered model type). Note also thatwe use (M� /p) to denote the time derivative of real balances M/p and M/p� for its rateof growth. � By way of k1 �K1 � n� sY/K� n� sf (k)/k� n (k1 � k� /k).


Define perceived disposable income of households by

Y��Y�M

p��T�TR,

where � is the expected (here equal to the actual) rate of inflation, T aretaxes and TR are transfers. Besides capital, the Tobin extension of theSolow model has one further asset, i.e. money, here considered in the formof real balancesM/p. The above concept of perceived disposable income isexactly equal to the rate at which private households can consume whileleaving their real wealth, defined by K�M/p, intact.� This is a well-known definition of perceived disposable income in the macroeconomicliterature and it leads to the formulation of the following consumptionfunction (where c is equal to 1� s):

C� cY�� c�Y�M

p� �T�TR� .

Private savings S�is then given by actual disposable income (Y��

Y�T�TR) minus consumption, i.e.

S��Y�T�TR� cY��K� �M� /p, (1.7)

and it is spent on capital formation and the increase in nominal balances asshown in the above flow budget restriction of households (1.7). Bymeans ofthe government budget restriction (1.5) this budget restriction can befurther reformulated as

Y�M� /p� c�Y�M

p��M� /p��K� �M� /p,

or

(1� c)�Y�M

p� �M� /p��K� �M� /p�

M

p�.

Due to � � p� /p and (M� /p)�M� /p�Mp� /p��M� /p� (M/p)(p� /p) this lastequation then gives rise to equation (1.1). We therefore see that equation(1.1) is more than just a description of private savings behavior.Equation (1.2) adds a simple LM equation to this description of asset

accumulation where the nominal rate of interest is given by the rate ofprofit � �F

�� (Y� �L)/K plus the expected rate of inflation �. Note

that here the expected rate of inflation is equal to the actual rate of inflationsince the present formulation of the model is based on the assumption (1.6)of myopic perfect foresight. Equations (1.3) and (1.4) are the conventional

� See Sargent (1987, pp.18—19) for details.


equations of neoclassical full employment growth (based on the neoclassi-cal theory of income distribution). Equations (1.5) and (1.6) have alreadybeen explained above.Comparing this model of monetary growth with the Solow growth

model we thus find that its new features are,

• a capital accumulation equation�� which is based on a new concept ofperceived disposable income and a simple, but consistent governmentbudget equation;

• a description of money-market behavior;• the assumption of myopic perfect foresight with respect to the rate ofinflation, where inflation, as a new feature, is generated by the growth inmoney supply.

Instead of the crude version of Say’s Law employed in the Solow model(I�S� sY) we have a more refined version of this law in the presentmodel which follows from the two relationships (see above)S��Y�T�TR�C�K� �M� /p, and S

��T�TR��M� /p, i.e.

S��S

�� S�Y�C�K� , which states that all output not consumed will

in fact be invested, since any investment in real balances must be equal tothe difference between actual disposable income and output if it is assumed,as is the case here, that households absorb the increase in money supplyM� .��The above model is, as is Solow’s growth model, a model of full tempor-

ary equilibrium (on the markets for goods, labor, and money), which,however, is based on Say’s Law and not on some sort of Walras’ Law as issometimes believed to be the case. This is of importance when consideringand interpreting its conventional extensions where money market disequi-librium is considered side by side with equilibrium in the market for bothgoods and for labor.In intensive form, the above model gives rise to (k� K/L,x�Y/

L� f (k),m�M/(pL)),�

k� � sf (k)� nk� (1� s)m(� p), (1.8)

m� � p� n, (1.9)

where p is obtained from (see equations (1.1), (1.2), and (1.6)) m� h( f �� K� �Y�C�S� S

��S

�, see the following.

�� This formulation of Say’s Law implies that the goods market will always be in equilibrium,quite independently of the state of themarket formoney or for labor. Such a situation is notpossible in models that rely on some form of Walras’ Law.

� If Y��Y��Y�T�TR is assumed instead of the above we would obtain here:k� � sf (k)� nk� (1� s)m.


(k)� p) f (k), by making use of k� � (K1 � n)k�K� /L� nk and equation(1.1), since (M� /p)� (� p)M/p.The model therefore exhibits a new term in its fundamental equation,

namely � (1� s)m(� p), which makes capital accumulation now alsodependent on real balances per head and money market phenomena.Furthermore, it now consists of two dynamical laws (for k andm) instead ofthe single law (for k) of the Solow model.The steady state of the model (where � p� n holds) is described by

0� sf (k)� nk� (1� s)nmm� h( f �(k)� � n) f (k).

Assuming that there is a unique solution to these equations�� it followsimmediately by total differentiation ( f � � 0, f �� 0, h�� 0) that dk/d� 0must hold true, which is the so-called Tobin effect. This intuitively plaus-ible effect states that if the return of holding money as an asset is reduced(by increasing the steady state rate of inflation �M� /M), the relativecomposition of assets will shift toward capital, increasing thereby capitalintensity k and output per head y� f (k). Instead of the superneutrality ofmoney (where does not influence the real part of the economy in thesteady state), we thus have a positive influence of the growth rate of moneysupply on capital, output, and consumption per head.This steady-state result has been extended, turned into a negative Tobin-

effect and also shown to be non-existent through a variety of modificationsof the original Tobin model in descriptive as well as optimizing macro-economic frameworks. This literature is surveyed in Orphanides andSolow (1990) and will not be considered here, since we only want to give abrief survey on the dynamic properties of the Tobin model and its exten-sion to adaptive expectations in this section.Nagatani (1970) has investigated the stability of the Tobin model under

the assumed situation of myopic perfect foresight and found that it exhibitssaddlepoint instability. As Orphanides and Solow (1990) state: ‘‘At thistime this was considered to be a fatal flaw of the model, . . .’’ Thus itappeared at that time that the steady-state analysis of Tobin was notsupported by dynamic analysis as far as the myopic perfect foresight case(1.6) was concerned.Yet, Sidrauski (1967a) had already shown that if inflationary expecta-

tions � are formed adaptively and at a sufficiently slow rate, then the Tobinmodel was globally asymptotically stable. This modification, instead of(1.8), (1.9) gives rise to

�� Which can easily be shown by means of simple assumptions on f and h, since the functionnk/f (k)� (1� s)nh( f �(k)� �n) involved in the determination of the steady state isstrictly increasing.


k� � sf (k)� nk� (1� s)m(� �),�� (p� �)��(� n�� m), (�� 0),

where m is given by m� h( f �(k)� �) f (k), which would therefore supportthe steady-state arguments of Tobin again.�� To explore the stabilityproperties of this steady-state solution further it was also often assumed, inparticular by Hadjimichalakis (1971a,b),�� that prices are not always equi-librating, but are responding sluggishly to money market disequilibriumand, due to Fischer’s (1972) proposal, also dependent on the expected rateof inflation �, as for example in the following determination of the rate ofinflation:

p��(m� h( f �(k)� �) f (k))��.

The dynamic system thereby becomes a three-dimensional one and of theform

k� � sf (k)� nk� (1� s)m(� �),m� �m(� p� n)�m(� n��

�(m� h( f �(k)� �) f (k))),

�� (p� �)��(m� h( f �(k)� �) f (k)).

The advantage of this formulation of the dynamics around Tobin’s steadystate is that its feedback structure is more easily understood (or moredisentangled), but at the cost of increasing the dimension of the dynamicsby one and by a perhaps not widely accepted adjustment rule for the pricelevel p. Be that as it may, this generalization of the dynamics (andmodifica-tions of it) have been extensively studied in the literature.Hayakawa (1984),for example, uses a general flow disequilibrium concept in the market formoney in order to systematize the local stability properties of variousapproaches to the determination of p by means of special cases of it.Benhabib and Miyao (1981) show that the generalized Tobin model losesstability in a cyclical fashion by way of a Hopf bifurcation if the parameter�� of the expectations mechanism becomes sufficiently large. The generalimpression that arises from these treatments of the stability problems of theTobin model is that adaptive expectations support instability if they areformed with sufficient strength (with a lag that is sufficiently short), whiletheir limit case �� of myopic perfect foresight directly gives instabilityin the form of a (local) saddlepath dynamics.In our view, the basic explanation for such a result is that there is a

positive feedback mechanism in this model and its variants, of the form

��m� h( f �(k)��( f (k))�p�� ,

which becomes stronger the larger the adjustment speeds ��, �� of prices

�� The steady state of this model is the same as that of the Tobin model (1.1)—(1.6).�� See Sijben (1977) for an early systematic treatment of the literature on the Tobin model.


and of inflationary expectations become. This mechanism is much moregeneral than its present appearance in generalized Tobin models and willreappear in various forms throughout this book. The value of descriptivegeneralized Tobin models is that investigation of the above positive feed-back mechanism within them has been extensive, while it has been largelyneglected so far in models of the Keynesian variety. It is the Tobin modeltype where the role of the adjustment parameters �� and �

�has been

investigated the most thoroughly.��Summarizing, we can state that the neoclassical monetary growthmodel

of Tobin (1965) type integrates portfolio choices and their implications forthe determination of perceived disposable income into the standard des-criptive neoclassical approach to real growth. It is based on Say’s Law(sometimes confused with some sort of Walras’ Law) and full employmentof the labor force (which is not a compelling assumption, as we shall see inchapter 2). It treats the role of the government in a very simple way(confined to the supply of money which is, however, related to, and thus notindependent of, fiscal policy). Many later versions of the Tobin model haveassumed various forms of moneymarket disequilibrium to allow for a finiteadjustment speed of the price level (some sort of a crude but dynamicquantity theory of money), combined with adaptively formed expectationsor myopic perfect foresight.The validity of the Tobin effect, the denial of the superneutrality of

money in particular, was also considered by means of models which putmoney in the utility or the production function. Following an early proof ofsuperneutrality by Sidrauski (1967b) in an optimizing framework most ofthe discussions of money in a growing economy are now based on such amicro founded approach, assuming infinitely lived individuals, overlap-ping generations or cash-in-advance situations (see Orphanides and Solow1990 for a detailed presentation of these approaches). The Tobin (1965)model thereby becomes a truly neoclassical one, with optimizing agentsand general equilibrium as for example in Gale (1983, ch. 2).Orphanides and Solow (1990, p. 225) state in this regard:

The main lessons were thus already implicit in the work of Tobin and Sidrauski.For those who can bring themselves to accept the single-consumer, infinite-horizon, maximization model as a reasonable approximation to economic life,superneutrality is a defensible presumption. All others have to be ready for adifferent outcome.

An important further development that grew out of the discussion of

�� We have already cited Hayakawa (1984). See also Chiarella (1986) and Chiarella andLorenz (1996) for a nonlinear approach whichmakes the locally explosive dynamics of thismodel type globally viable without reference to the so-called jump-variable technique.


Tobin’s approach to monetary growth was given by the treatment ofmyopic perfect foresight that it eventually induced. After recognizing thesaddlepath instability of perfect foresight solutions (and the instability offast adaptive expectations) most authors in monetary growth theoryadapted a strategy to cope with the former problem, as it was proposed bySargent and Wallace (1973), that turned saddlepath instability into so-called saddlepath stability by the choice of a certain jump-variable tech-nique. This redefinition in the dynamic treatment of (local) saddlepathsituations (and the devaluation of the adaptive expectations mechanism)has implied that the stability problems of the Tobin model and its exten-sions became irrelevant in the further discussion of monetary growth.Therefore, there has never been an attempt in the orthodox literature toovercome these related local instability scenarios by an alternative reactionto them, which assumes appropriate nonlinearities far off the steady statein order to keep the observed saddlepath dynamics of the model economi-cally meaningful for all points in the phase space from the global point ofview. Chiarella (1986, 1990) and Flaschel and Sethi (1999) develop such anapproach, the essence of which is outlined in section 1.6 below. Such areaction to local instability may produce interesting cyclical growth pro-cesses, not only for the Tobin model but also for other, less neoclassicalapproaches to monetary growth, as we shall see on various occasionsthroughout this book.

1.4 Basic Keynes–Wicksell models of monetary growth

Monetary growth models of Keynes—Wicksell type were developed in thelate sixties and early seventies, in part as a reaction to the Tobin neoclassi-cal model of monetary growth. They assume the existence of an investmentbehavior that is formulated independently of savings behavior, and thus nolonger rely on some sort of Say’s Law.�� The instability problems of theTobin growth model can also be established for the models of Keynes—Wicksell type, but have somehow been neglected here, since fast adaptiveexpectations, or even myopic perfect foresight, has not been a topic in theseapproaches to monetary growth dynamics.� Models of Keynes—Wickselltype are now almost forgotten in the discussion of monetary growth.However such neglect is not justified as these models represent a usefulgeneralization of what is now known as textbook Keynesian dynamics, aswe shall see in Chapters 3 and 5. We refer the reader to Orphanides and

�� Though the version of Walras’ Law these models assume is often not made explicit.� See Stein (1982) for example. Fischer (1972) offers such an analysis, but is not fully correct in

his findings on the local stability of the model, since he uses an incorrect expression for thetrace of the Jacobian of his dynamical system at the steady state.


Solow (1990) for a brief characterization of this model type. Rose (1990)presents an isolated, but very interesting, attempt to make this model typea more consistent and general one in such a way that many other ap-proaches to economic dynamics can be treated as special cases of it.Models of Keynes—Wicksell type have been developed by Rose (1966,

1967, 1969) and Stein (1966, 1968, 1970, 1971) in particular and as analternative to neoclassical models of monetary growth. Instead of moneymarket disequilibrium, they assume goods and labor market disequilib-rium, while keeping the money market always in equilibrium. In our view,this is a more plausible disequilibrium scenario than that of the Tobinmodels with their pre-Keynesian structure. However, Keynes—Wicksellmodels do not addmuch to the steady-state properties of the Tobinmodels.Their value must therefore be sought in the more plausible and interestingdynamical features to which they are supposed to give rise.In this respect, the model developed by Rose (1967) is the most interest-

ing one. It is kept two-dimensional by simplifying the money-market andinterest-rate determination appropriately, whereas Keynes—Wicksellmodels of monetary growth in general need at least three dynamical lawswhen they are set up as complete models of monetary growth (see Fischer1972 in this regard).� Keeping the dynamics two-dimensional has theadvantage that global stability results can be obtained more easily than inthe third dimension by assuming certain basic types of nonlinearities, andthis is indeed what makes Rose’s (1967) approach interesting even fromtoday’s perspective, where nonlinearities in economic behavior are receiv-ing renewed attention.As does the Tobin model considered in the preceding section, the Rose

model assumes neoclassical smooth factor substitution, which is hererepresented by the following intensive form production function

f (l)� y, l�L/K, y�Y/K, f �� 0, f �� 0, (1.10)

instead of making use of k�K/L and x�Y/L as in the Solow—Tobinmodel. Though Rose also considers monopolistic competition, here weshall make use only of the competitive limit case of this theory of priceformation. Thus, in addition to (1.10), we again assume (as in the Tobinmodel) the validity of the marginal productivity rule,

��w/p� f �(l). (1.11)

Money-market influences are trivialized in the Rose model by assumingthat the rate of interest r is an increasing function of l, the labor intensity��

� The dynamic variables then in general are ��w/p, m�M/(pL) and �.�� In the Tobin model (see equation (1.2) of the preceding section), we instead haver[� ��]� h��(m/y),m�M/(pK), i.e., r depends on l and m there.


(see Rose 1967, p. 171, for a justification of this short-cut, which is based onan endogenous money supply rule in particular).Generalizing the simple savings function sy� sf (l) of the Solow model,

Rose assumes it to be of the form

s(l)� g(l, r(l)), s�(l)� 0, (1.12)

and for investment he assumes

i� i(l), i�(l)� 0, (1.13)

which he rationalizes by stating that i is a strictly increasing function of therate of profit � of the model which in this context is given by

�� f (l)� f �(l)l��(l),��(l)� � f �(l)l� 0.

Note here, that both s and i are already calculated per unit of the capitalstockK, the same as l and y. Note also that the rate of interest and the rateof inflation play no role in determining investment behavior (the latter alsodoes not appear in the description of savings behavior). Investment (andsaving) are thus determined solely by real magnitudes.Since we have assumed perfect competition (or �� in the notation of

Rose 1967, p. 166) we get from his general model the following specialdetermination of price inflation

p�H(i(l)� s(l)),H�� 0,H(0)� 0, (1.14)

which states that the rate of inflation is an increasing function of thedemand gap (I�S)/K on the market for goods, an important ingredient ofmodels of Keynes—Wicksell type. Similarly, wage inflation is determined bythe demand gap in the market for labor (V�L/L,V� � 1 the full utiliz-ation rate of the labor force),

w�G(V ),G�� 0,G(V� )� 0 for V� � (0, 1), (1.15)

where the function G is assumed to become steeper and steeper the fartherthe rate of employment departs from its fundamental position V� .�� Insimple economic terms, Rose’s model assumes that price flexibility remainslimited, while wage flexibility becomes unbounded at a certain distancefrom its fundamental position V� .The remaining assumptions of the Rose (1967) model are

K1 � s(l), (1.16)

L1 � n, (1.17)

i.e., they describe the growth of the productive factors in the usual way.

�� Which need not be the steady state position in the Rose (1967) model.


This model gives rise to the following two dynamical laws (��w/p thereal wage):

��G(l/l)�H� (l),H� (l)�H(i(l)� s(l)), (1.18)

l1 � n� s(l), (1.19)

where l is a strictly decreasing function of �, namely, l� ( f �)��(�) (seeequation (1.11)).� It can therefore be reduced to only two — nonlinear —laws of motion for the real wage � and for the fullemployment laborintensity l. Real wage dynamics are here driven by both excess demand inthe labor market (a positive dependence) and excess demand in the marketfor goods (a negative dependence). This is a very interesting feature of thisversion of a Keynes—Wicksell model, since it is often the case in theliterature (even today) that only the former disequilibrium enters theequation of real wage dynamics. Accumulation in equation (1.19), on theother hand, is described in a way that is customary for models of the Solowtype. We note here that prices as well as wages are predetermined at eachmoment in time. Causality therefore runs from a given real wage to thethereby determined level of employment L� l(�)K, which is unusual fora Keynesian model of monetary growth.Rose (1967) shows that the abovemodel has a unique steady state, which

is locally unstable ifH� � is positive�� and relatively large compared to G� atthe steady state, but which is globally stable if H� , the measure of priceflexibility, remains bounded for all values of l, while the slope of G,measuring wage flexibility, approaches infinity at some distance from thesteady state. This gives rise to a unique limit cycle or persistent cyclicalmotion in real wages and full-employment labor intensity by means of asuitable application of the Poincare—Bendixson theorem for planar dy-namical systems.This is the employment cycle result of Rose (1967), which is derived there

in a very detailed way and which is explained by him in economic terms inhis section 6. We shall return to this employment cycle model in chapter 5of this book when smooth factor substitution is added to the prototypemodels considered in chapters 2—4.The locally destabilizing force of the model is a high degree of price

flexibility (relative to wage flexibility at the steady state), combined with asavings and investment behavior in which investment is more sensitive toreal wage changes than savings. These assumptions guarantee that thestabilizing influence of the Phillips curveG on the numerator of � �w/p isovercome by the destabilizing influence of H� on the denominator of

� Rose makes use of the dynamic variable l in place of � which makes the necessarycalculations less straightforward. �� i�(l)� s�(l) and H� � 0 and large.


��w/p. An increase in real wages decreases goods market excess demandby so much that prices will fall faster than wages, which in turn means thatreal wages are further increased, etc. Note that this description only holdsin the neighborhood of the steady state, since wages become more andmore flexible the further the economy deviates from its steady state path.Far off the steady state, nominal wages have become so flexible, that realwage increases now exercise a negative influence on the time rate of changeof real wages, since money wages now fall faster than the price level. As wehave indicated above, the model thereby becomes a globally stable one,which in conjunction with the local instability produces the employmentcycle that is the theme of Rose’s (1967) paper.Comparing Rose (1967) with Goodwin’s (1967) famous growth cycle

model furthermore suggests that the Goodwin profit squeeze mechanism isalso the basic driving force in the Rose employment cycle result, but it ismodified there in two ways. First, adding substitution to the model ingeneral makes the Goodwin growth cycle converge to the steady state, i.e.,it makes this model a globally asymptotically stable one. Secondly, addingthe destabilizing price mechanism to the model (see above) prevents con-vergence to the steady state, but nevertheless ensures global stability if thewage mechanism is stronger than the price mechanism far off the steadystate. The limit cycle result of Rose is therefore due to the stabilizing role ofsmooth factor substitution (combined with money wage flexibility) and tothe destabilizing inflationary Wicksell mechanism both superimposed onthe Classical growth cycle model of Goodwin.�� This questions to someextent the label ‘‘Keynes—Wicksell’’ given to this model type, since thestructural equations and their implications in fact appear to be moreClassical than Keynesian in nature. We shall see in chapter 3 of this bookthat this observation holds true also in the case of the inclusion of a properLM equation into the model, due to the Wicksellian use of IS-disequilib-rium that is characteristic for all models of the Keynes—Wicksell type.We add without demonstration that the verbal description of real wage,

nominal wage, and price dynamics can be fairly complicated despite thesimple structural equations of the Rose model (see also the descriptiongiven in Rose 1967, p. 165, and note that the one given in Ferri andGreenberg 1989, p.51, is much too simplified). This is due to the manysituations of falling and rising nominal values that can be observed in thecourse of this employment cycle. Note, finally, that we have equality ofwage and price inflation in the steady state �

�.

The Rose model thus differs from the Goodwin model by its moregeneral real wage dynamics, which is backed up by a separate description

�� The Goodwin growth cycle is approached by the Rose cycle for low elasticities ofsubstitution in production and a flat H( · ) schedule.


of nominal wage and price dynamics. This extension makes it possible forthe Goodwin overshooting mechanism to be now represented by (oftenonly) one closed orbit which is attracting all other trajectories.�Keynes—Wicksell models exhibit in general an investment function that

is more general than that of Rose (by including in particular interestpayments as a cost of financing investment) and they explain movements inthe rate of interest by means of a properly specified LM equation. Such anextensionmay eliminate the limit cycle from the above model in the case ofa very flexible interest rate (see chapter 3). The really crucial problem forRose’s approach is, however, given by another of its omissions, namely, theaccelerator expression �� (the expected rate of inflation), in its formula-tion of the money wage Phillips curve mechanism�

w�G(l(�)/l)� �.

This omission of � in the money wage dynamics is motivated by Rose(1967, p. 167) by assuming that � is a function of l:�(l),�� 0 which doesnot change the qualitative features of equation (1.15) and of the whole Rosemodel.Yet, assuming the above equation in place of equation (1.15)modifies the

model significantly, as we now need to specify a mechanismwhich explainsthe formation of inflationary expectations �. Doing this as in the precedingsection on the Tobinmodel bymeans of �� (p� �), (��) then leadsto either three-dimensional dynamics (if �� ) or to a real wage Phillipscurve � �G(l(�)/l) (if �� : p��). The latter case of myopic perfectforesight therefore removes the destabilizing goods-market component �from the model and gives rise to a Goodwin growth cycle model withsmooth factor substitution, which is known to be globally asymptoticallystable, see Flaschel (1993, ch. 4).Adding to the model the above mentioned extended investment function

i(�(�)� (r� �))� n, (i�� 0), which refers to the real rate of interest as theappropriate determination of the costs of investment (and which has atrend term n equal to natural growth to avoid problems of steady statedetermination), and adding a theory of interest as in the Tobinmodel, i.e., aKeynesian LM equation:M� pYh(r), h�� 0, then gives rise to the monet-ary Keynes—Wicksell model as it was used in the literature after theappearance of Fischer’s (1972) critique of the structural equations and thesteadystate properties of Keynes—Wicksell models. Such an extension ofRose’s Keynes—Wicksell employment cycle model has local stability prop-erties that are close to those of the generalized Tobin model considered in

� We shall see in chapter 3 that Rose’s assumption of smooth factor substitution is notimportant in this respect.

� Rose’s result would stay intact, if one uses ��, �� 1 in the place of ��!


the preceding section. As already stated, the local stability properties ofKeynes—Wicksell models have not been carefully investigated in the litera-ture whenever their dynamical dimension becomes larger than two,� sothat their intimate relationship to generalized models of the Tobin type isnot well-documented. Furthermore, there has been no attempt to performa global stability analysis as in Rose (1967) for a higher dimensionalKeynes—Wicksell dynamics. Interesting as the nonlinear analysis of thebusiness cycle of Rose (1967) is in itself, it has therefore never been treatedon a broader basis. Such a treatment is called for by models which trulyintegrate monetary growth and the formation of inflationary expectations insuch an environment by means of the above LM equation, the aboveextended investment function and the Tobin effect in the savings function.Summarizing, we may state that monetary growth models of Keynes—

Wicksell type break with Say’s Law (as they should), and establish somesort of Walras’ Law by which financial assets other than money areexcluded from explicit consideration (see chapter 3 for more detailed andmore recent formulations). They assume LM-equilibrium as in the Tobinmonetary growth model, but depart from this model by assuming a profitrate/real rate of interest-differential that drives investment decisions (seealso Sargent 1987, pp.11, 83, on this point), which are then separated fromsavings decisions. Keynes—Wicksell models have a somewhat elaboratedwage—price sector where both wages and prices are driven by correspond-ing demand pressure and sometimes also by appropriate cost-push termsor certain expectations about them (see Flaschel 1993, Flaschel and Sethi1996, for a general treatment of such extended approaches). We shallprovide in chapter 3 an even more general treatment of the Keynes—Wick-sell prototype model which integrates Rose’s (1967) nonlinear treatmentwith newer formulations of the wage price sector of Rose (1990) which weshall briefly consider below.Above we have criticized Rose’s (1967) formulation of the wage—price

sector as being too limited with respect to cost-push terms and expecta-tions about them. In a recent book, Rose (1990) has provided an interestingextension of this wage—price sector which by and large meets this criticismwithout losing the innovation his 1967 model where the real wage dynam-ics depend on both themarket for labor and themarket for goods. This newformulation of wage—price dynamics is described by the following pair ofadjustment equations or Phillips curves,

� See Stein (1982) for a relatively recent, and fairly general, model of Keynes—Wicksell typefor which no stability analysis is provided, even from the local point of view.

See also Asada (1991) for an alternative extension of the wage price sector of a model ofKeynes—Wicksell type.

Slightly modified to serve our purposes in the following chapters.


w�G(l/l)� �p� (1�

�)�,

p �H(i(l)� s(l))� �w� (1�

�)�,

where G andH are given as before, and where p, w denote actual price andwage inflation and � a medium-run expected rate of inflation. The par-ameters

�, �are in the interval [0, 1], where the case

��

�� 1 is

generally excluded, since it would give rise to two independent dynamicallaws for real wage determination. If this case is excluded, we can reformu-late the above wage—price sector in the followingway and then solve for thetwo variables w��, p��:

w� ��G(l/l)� �(p��),

p�� H(i(l)� s(l))� �(w��),

implying ( � (1� � �)��),

w� �� [G(l/l)� �H(i(l)� s(l))],

p�� [H(i(l)� s(l))� �G(l/l)],

and therefore

�� w� p� [(1� �)G(l/l)� (

�� 1)H(i(l)� s(l))].

We therefore get a real wage dynamics of the same type as in the originalRose model, but based now on myopic perfect foresight with respect to wand p, and on a variable � which represents the state of medium-terminflationary expectations. This extension of the wage—price sector will beused in our reconsiderations of the Tobin, Keynes—Wicksell, and Key-nesian prototype models and represents, in our view, the really innovativecontribution of models of the Keynes—Wicksell type.Note here also that the original Rose employment cycle determined the

steady-state value of the real wage ��by means of n� s(l(�

�)), which

implies that we get for wage and price inflation

w�G(l(��)/l)�H� (l(�

�))� p,

a value that is in general not zero, i.e., we may, for example, have a positiverate of inflation and also i(l(�

�))� s(l(�

�) in the steady state. The fact that

the steady-state concept of Keynes—Wicksellmodels allows for I� S in thesteady state has been criticized by Fischer (1972), and has led to theinclusion of the accelerator term �� in the formulation of wage—pricedynamics. The above reformulation of the wage—price dynamics as in Rose(1990) is a different, or more general, answer to the problem that steady-state inflation should be compatible with the I�S condition, since it iseasily seen from the above that w� p� � will imply G(l/l)� 0 andH(i(l)� s(l))� 0 if

� �� 1 holds. This approach therefore also allows


for a deviation of money supply growth from natural growth n withoutimplying goods-market disequilibrium in such a case.In sum, Keynes—Wicksell models have thus led to a reformulation of

wage—price dynamics which favors the use of two symmetrically builtPhillips curves instead of the one of the Samuelson—Solow (1960) typewhich has dominated the discussion of wage and price inflation (aug-mented by inflationary expectations) since the invention of the Phillipscurve.

1.5 Basic AS–AD growth models

1.5.1 The so-called ‘‘Classical’’ model and its dynamics

Today’s macroeconomic textbooks generally distinguish between the so-called ‘‘Classical’’ and the ‘‘Keynesian’’ model of a monetary economy (seeMcCallum 1989 for example). By this distinction is meant the model of theneoclassical synthesis of Keynesian economics which assumes perfectlyflexible money wages in the first case and assumes (temporarily) givenmoney wages in the second case. Perhaps the best presentation of thisdistinction is given in Sargent (1987, chs. 1, 2). We shall use his presentationin this and the next subsection to briefly present fully dynamic versions ofhis two (mainly static) analyses of the Classical and the Keynesianmodel ofthe neoclassical synthesis (see also Sargent 1987, p.xvii, in this regard). Ourdynamic versions nevertheless simply analyze what is already contained inSargent’s (1987) chapters 1, 2, and 5 by putting together the dynamicelements he proposes at various occasions in his book. In this way, com-plete models of this Classical and Keynesian prototype are obtained whichcan then be judgedwith respect to their implications and their true general-ity. Our findings will be that these twomodel types are much too narrow intheir assumptions to allow for a meaningful comparison of (neo-)ClassicalandKeynesian dynamics (see chapters 2, 3 and 4 in this volume), even whenit is admitted that asset markets and investment remain, as in Sargent(1987, ch. 1), specified in the neoclassical sense. In particular in the assetmarkets money is then held solely for transaction purposes, bonds are likesavings deposits, equities are identified with these bonds via the perfect-substitutability assumption, and investment is of the Wicksellian type (seethe preceding section), with trend term nK, guaranteeing a full employmentsteady state.The crucial lacking element, one that distinguishes Classical from Key-

nesian versions, but which is lacking in all presentations of this distinction,is that labor and capital will be underutilized (or overemployed) duringcertain phases of the cycle. Or, to express it in an equivalent way, not only


wages but also the price level adjust with finite speed. Neglecting thistwofold disequilibrium situation produces a considerable bias in the Key-nesian version of the ‘‘neoclassical synthesis’’ toward neoclassical results, aswe shall show in chapters 3 and 4 (for fixed proportions), and in chapter 5(for smooth factor substitution).The contrast between the Classical and the Keynesian views on macro-

dynamics therefore only becomes really obvious (at a very elementary stagestill, as we have seen in the preceding section), when there is symmetry inthe assumption on wage and price flexibility in both the Classical (bothinfinitely flexible) and the Keynesian models (both less than infinitelyflexible). By and large, this simply amounts to accepting the existence of asecond Phillips curve, besides Phillips’ nominal wage inflation curve —namely, one for the price level, which is to some extent independent ofmoney wage dynamics and possibly also differently structured. This simpleand plausible statement will lead to many new aspects in neoclassical andKeynesian macrodynamics, as we shall see in the body of this book. Butbefore this task is undertaken, we have to sketch the state of the art informulating the Classical and Keynesian models of the neoclassical syn-thesis which provide the dominant view on this distinction. We do this bymaking them truly models of monetary growth, so that they can be indeedcompared to our subsequent monetary growth models.The (completed) Classical model of Sargent (1987, ch. 1) consists of the

following set of equations (see his p.20),

Y�F(K,L),w/p�F�(K,L),L�L, (1.20)

C�C�Y�T� �K�M�B

p� � qI, r� �� , (1.21)

I � i(q� 1)K� nK, q� (F�

� �� (r� �))/(r��), (1.22)

Y�C� I� �K�G, (1.23)

M� pYh(r),M1 �� const., (1.24)

L1 �� n,K1 � i( · )� n� I(q� 1)/K, (1.25)

�� (p��), 0��. (1.26)

Equations (1.20) represent the usual neoclassical presentation of the fullemployment position of the labor market, including full employment out-put.� Equations (1.21) and (1.22) describe consumption and investmentbehavior based on an advanced notion of disposable income Y� and

� To simplify the analysis of the dynamics analysis we have assumed that L is given in eachmoment in time.


Tobin’s q. We shall follow Sargent’s (1987, ch. 5) dynamic analysis of theKeynesian model in the following and conduct such dynamic analysisunder the simplifying assumption C� c · (Y� �K�T) (and I� i · (F

��

�� (r��))K� nK), and thus shall exclude Tobin effects (see section 1.2)and real interest or Mundell effects in the consumption function (i.e., wetakeC�� 0,C

�� 0). Equation (1.23) is the goods-market equilibrium con-

dition and equation (1.24) describes (again) the money-market equilibrium.Equations (1.25) describe factor growth as determined by the exogenouslabor supply growing at the rate n. Equation (1.26) finally explains infla-tionary expectations (either adaptive expectations when �� ormyopicperfect foresight when ��).The intensive form dynamics of the above model reads (L/K� l),

y � f (l),�� f �(l),�(�)� f (l)��l, y�Y/K etc. (1.27)

y � c(y� t� �)� i(�(�)� r��)� �� n� g, (1.28)

m� yh(r),m�M/(pK), (1.29)

l1 �� i( · ), (1.30)

�� (� n� m� i( · )��), (1.31)

where t�T/K, g�G/K are assumed constant (just as n, �, see Sargent1987, ch. 5). The last two equations describe the dynamics of the model,while the second and the third determine in an IS—LM fashion the tempor-ary equilibrium variables m and r as functions of the dynamically en-dogenous variables l,� (which are statically exogenous). Note that y,� and� are all obviously determined by the given l (at time t), so that the supplyside of the model is very simple to calculate at each moment in time.Equation (1.28) can be easily solved for the nominal rate of interest and

gives

r ��c(y� � � t)� i(��)� n� � � g� y

i, y� f (l),�� (l),

which implies r� r(l,�), r�� 1. The other statically endogenous variablemis then simply given bym �� f (l)h(r)�m(l,�) withm� � f (l)h�(r)r�� f (l)h�(r).For the rate of growth of the variable m we thereby get

m�m�ml� �

m�m· ��

m�l

ml1 �

h�(r)h(r)

· �� ,

an equation which can be used to remove the m expression from the twodynamical laws for l and �.Let us now consider in turn the cases of adaptive expectations and


myopic perfect foresight. In the first situation we get from the abovedynamical system,

l1 �� i(�(l)� r(l,�)��), �(l)� f (l)� f �(l)l� n� (1� c) f (l)� c(t� �)� � � g, (1.32)

�� 1

1��h�(r)/h(r)�� n��

m�l

m� 1� l1 �� . (1.33)

The second form of equation (1.32) immediately implies a globally asymp-totically stable monotonic adjustment of the variable l to its steady-statevalue l

�, given by

l�� f��((n� � � g� c(t� �))/(1� c)),

which is completely independent of the � dynamics. The partial derivativeof �� with respect to � at the steady state, on the other hand, is given by

�� 1

1/�� h�(r�)/h(r

�), r

�� r(l

�, �

�),�

�� n.

It is therefore negative for all �� h(r�)/h�(r

�), and positive for all

�� h(r�)/h�(r

�) (and ill-defined in between).

Shocks � in the growth rate of the money supplyM therefore leavethe real part of the steady-state solution unchanged, while the expected rateof inflation converges to the new steady-state value of the rate of inflationin the first of the above cases and diverges from it towards�� accordingto �� 0,�0. This is all that happens in this monetary growth model inresponse to changes in the growth rate of money supply. Of course, shocksto the monetary part of the economy are somewhat more difficult toanalyze, as one then has to treat the nonautonomous system

�� d(l(t), �),

where d is the function on the right hand side of equation (1.33) and wherel(t) is given by the solution of (1.32).This, however, does not prevent the general result that the steady state of

the dynamical system (1.32), (1.33) is a saddlepoint for all adjustmentspeeds of the adaptive expectations mechanism that are chosen sufficientlyhigh. As an economic model these dynamics are therefore incompletelyspecified in the case of fast adaptive expectations and hence the modelneeds further assumptions on its behavior far off the steady state. Suchassumptions are, however, not discussed by the authors who consider theClassical version of the neoclassical synthesis to be a meaningful descrip-tion of economic states and dynamics. Furthermore, even when this dy-namic model is made an economically meaningful (that is to say a viable)


one,� its dichotomizing dynamics will be of a very restrictive nature, thedescriptive value of which may be very much doubted.But advocates of such Classical models usually do not favor the assump-

tion of adaptive expectations, so they may escape from the above con-clusions by assuming the much more favored situation of myopic perfectforesight in the place of this purely backward-looking expectations mech-anism. Let us therefore consider also this second case of expectationsformation.In this case, one has, because of �� p, the following new relationships

which determine now the dynamics ofm by describing a locus of admissiblem� —m combinations.

y(l)� c(y(l)� �� t)� i(�(l)� h��(m/y(l))� � n� m� l1 )� n� �� g,

where l1 is again given by

l1 � n� (1� c) f (l)� c(t� �)� �� g,

i.e., a Solovian description of the process of real growth that is independentof the development of monetary variables. The above determination of them� �m relationship implies a functional form for this relationship of thefollowing type:

m� e(m, l), e�� 0, e

�� 0.

When this last dynamical law is taken together with the l dynamics (givenabove), we again obtain a dichotomizing system with an independentSolovian real growth process and a saddlepath situation (which impliescomplete instability, as in the case of fast adaptive expectations) for themonetary sector, if the above dynamics are solved by means of historicallygiven values for the monetary variablem. Since Sargent andWallace (1973,see also Sargent 1987, I.9), it has however become the fashion to solve sucha dynamical law in a purely forward-lookingway bymeans of an appropri-ate terminal condition. This is known to work if the function h is a linearfunction and it makes m a jump-variable which jumps whenever such ajump is needed for the fulfillment of the terminal condition. In the case of ajump inM, however, no jump in m is needed, but just a jump in the pricelevel p (and w) such that m can remain constant, implying neutrality for theconsidered Classical monetary growth dynamics.Also, a jump � in will only modify the dynamical law for the variable

m in the following way,

m� e(m, l)� i�.� This means that the trajectories of the dynamics stay within economically meaningful

bounds. p� � n� m� l1 , l1 � n�K1 .


Such a jump will thus be superneutral (since there is no effect on k� as in theTobinmodel) due to the restrictive concept of disposable income here used.Therefore, the nominal magnitudes p and w together withm are but a fairlyarbitrary appendage to the Solovian growth model and the l1 -dynamicssurrounding its steady state. This may be a happy state of affairs from astrictly neoclassical point of view, but — in our view — it is mainly a quiteextreme and restricted type of dynamical system which is devoid of anyinteresting proposition on economic dynamics and monetary growth. Ad-vocates of the Classical approach to macrostatics and macrodynamicsshould be capable of formulating a richer dynamics than the one we havediscussed above, even in the deterministic case and on the ‘‘textbook’’ levelconsidered here.Sargent (1987, ch. 3) also considers the Tobin variant of the Classical

model, the purely Classical model where investment (and the money mar-ket) need not be adjusted to a given state of aggregate supply (such thatKeynesian IS—LM analysis is formally present), but where we again haveSay’s Law K� � I�S�Y� �K�C�G in its ‘‘real’’ form. In this case,bonds and real capital are perfect substitutes (and equities are no longerneeded), so that we then have

�(l)�� r, �(l)� f (l)� f �(l)l.

In this case, the m dynamics is even simpler and determined by

m� f (l)h(�(l)� � n� m� l1 ),

where l1 and l(t) are given as before (there is now no investment function andIS equation which must be fulfilled simultaneously). Solved for m, theabove equation reads,

m� � �(l)� (1� c) f (l)� c(t� �)� (�� g)� h��(m/ f (l)),

and it can be treated again in the Sargent and Wallace (1973) mannerdiscussed above. This may be considered the true Classical model, and itwould lead us back to Tobin’s analysis if more sophisticated concepts ofdisposable income were inserted into it as in section 3 of this chapter.

1.5.2 The so-called ‘‘Keynesian’’ model and its dynamics

The Keynesian dynamical system follows from the Classical model of thepreceding section in that it dispenses with the full employment assumption(for labor!) and thus allows for l�L/K to be different from l�L/K andfor a ‘‘sluggish’’ money wage adjustment of the form

w�H(l/l)��,H� � 0,H(1)� 0, (1.34)


where 1 is assumed to represent the so-called natural rate of employment(see Sargent 1987, chs. 2 and 5, for details).The dynamical system (1.27)—(1.31) is again valid as in section 1.4 up to

the modification that we now have l1 �� i( · ) (� l1 , which is not treated asa dynamic variable there or here). Furthermore, m is now to be defined byM/(wK), so that equation (1.29) now reads m� ( f (l)/f �(l))h(r). Finally, theabove Phillips curve (1.34) implies m� �H(l/l)� �� i( · ), so that weend up here with a system of the temporary equilibrium variables l, y�

f (l), and r(� � f �(l)), and the dynamically endogenous variables l,m and �.Note that l, r are functions of these state variables by the IS—LMequationsof the model and the production function y� f (l).This three-dimensional dynamical system, which again needs an explicit

law for the dynamics of p in the case of adaptive expectations (�� (p��),��), has been extensively studied by Franke (1992a). Since itis implicitly defined it gives rise to a rather complicated dynamical systemin the three variables l, m, and �. Just as the Classical dynamical system insection 1.3, it shares the characteristic of being ill defined at a certain valueof �� and gives rise to saddlepath dynamics as �� approaches �. Inbetween, as Franke (1992a) shows, there exists a certain value of �� wherethe system undergoes a Hopf bifurcation and loses its stability in a cyclicalfashion. We thus again end up with an incompletely specified economicmacrodynamic model when the adjustment speed of the adaptive expecta-tions mechanism passes a certain critical value. Sargent’s (1987, ch. 5)analysis of the model bypasses all these details completely, by concentrat-ing on the special case �� 0 in the main, where the system can be shownto be asymptotically stable (see Flaschel 1993, ch. 6).The general conclusion that follows from this state of the art (which has

not been improved since the 1979 first edition of Sargent’s 1987 book) isthat these Keynesian dynamics have been only poorly investigated up toFranke’s (1992a) contribution. This conclusion also extends to Tur-novsky’s (1977a, ch. 8) analysis of such models and the literature on suchdescriptive Keynesian models of macrodynamics which appeared there-after. We shall show in chapters 4 and 5 that this sort of Keynesiandynamics is indeed a bastard one, i.e., a mixture (limit case) between fullcapacity Keynes—Wicksell growth models and a properly specified modelof Keynesian dynamics (with under- or overutilized production capacitiesof firms). The implication of the foregoing is that, although we still lack aclear understanding of this model type, we do not really need such ananalysis in the first place. Instead, Keynes—Wicksell models as well asKeynesian ones should each first be further explored within their ownconsistent frameworks.We have so far commented only on Keynesian dynamics as in Sargent


(1987, ch. 5) in the case of adaptive expectations. The alternative situationof myopic perfect foresight is considered in Flaschel (1993, ch. 7) in greatdetail and can, for the purposes of the following, here be characterizedbriefly as follows.In the case of �� p, it is assumed in Sargent (1987, ch. 5) that the Phillips

curve w�H(l/l)� p indeed defines a real-wage Phillips curve:

��H(l/l),��w/p.

As in section 1.4 we furthermore have the dynamical equation

l1 � n� (1� c) f (l)� c(t� �)� �� g,

which, together with the marginal productivity rule f �(l)� �, then definesan autonomous real dynamics of the Solow—Goodwin growth (cycle) type.These (asymptotically stable) dynamics of the real part of the modelrepresent only a simple extension of the purely Solovian dynamics of theClassical model of the preceding section.As for the monetary part of the model we once again have the equation

y� y(l)� c(y(l)� �� t)� i(�(l)� h��(m/y(l))� � n� m� l1 )� n� �� g,

which is subject to the same reasoning as in the preceding section, i.e., againgives rise to a further (nonautonomous) law of motion for real balances mthat may (or may not) be subjected to the jump-variable technique of the‘‘rational expectations school.’’Under myopic perfect foresight, the Sargent (1987, ch. 5) version of

Keynesian dynamics, where both the price level and the wage level becomejump-variables as in the Classical variant of the model, thus is not reallydifferent from that of the Classical dynamics. It would therefore seem thatthe assumption of rational expectations blurs important distinctions be-tween various schools of economic thought. This topic is further pursued inthe following section.

1.6 The modeling of expectations

The modeling of expectations (of wage inflation, price inflation, exchangerate depreciation, etc.) plays a central role in macrodynamic modeling andcertainly will play such a role in the models of this book.� In fact we havealready seen that the dynamical behavior of the previously discussedmodels can be locally stable (unstable) for low (high) speeds of adjustment

� See alsoChiarella andFlaschel (1998f ) for further investigations of the expectational side ofmacrodynamic models, there extended to the treatment of open economies.


of adaptive expectations and can be of saddlepoint type when perfectforesight is assumed.For almost a quarter of a century formal analysis and modeling in

macroeconomics has been dominated by the rational expectations hypoth-esis. Whilst the precise meaning of the term ‘‘rational expectations’’ itselfprovokes a deal of discussion, we shall here use it in the sense defined byTurnovsky (1995). That is, that the prediction made by economic agents ofa variable of interest (e.g., rate of price inflation or rate of exchange ratedepreciation) be consistent with the predictions generated by the model theagents have in mind. In continuous time deterministic models the rationalexpectations hypothesis reduces to myopic perfect foresight. Our modelingof expectations will be at odds with this currently fashionable approach, sothe purpose of this section is to explain the rationale for the expectationalmechanisms we shall adopt.Our approach stems from our view that rational expectations modeling

suffers from two defects which have never been properly addressed by itsproponents. One defect is of a theoretical nature, the other is of an empiri-cal nature. In section 1.6.1 we discuss the theoretical difficulty whichinvolves the widely used jump-variable technique. In section 1.6.2 wereview the body of empirical evidence which we believe should causeeconomic theorists to question the blind application of traditional rationalexpectationsmodeling. In section 1.6.3 we outline in detail the rationale forour approach to expectations modeling in this book.

1.6.1 A critique of the jump-variable technique

As we have seen in previous sections, rational expectations (perfect myopicforesight) models usually display a saddlepoint instability. The jump-variable technique was introduced by Sargent and Wallace (1973) tocircumvent this troublesome feature. In essence they argued that the vari-able on which expectations of changes are being formed (e.g., price orexchange rates) must be able to jump discontinuously at a certain point intime in order to allow the economic variables of the system to reach a stablemanifold of the saddlepoint. Thereafter, the system dynamics would lead tothe equilibrium point of the model.Sargent andWallace expounded the jump-variable technique within the

context of the simple Cagan monetary model (again see Turnovsky, 1995,ch. 3). The technique was generalized by a number of authors to handlelarger models, as discussed in Buiter (1984). In larger models an issue ofnonuniqueness also arises since there may be more than one way to set theeconomic variables onto a stable manifold, depending upon the relation-ship between the number of stable and unstable roots and the number of


available jump-variables. These issues are discussed by Blanchard andKahn (1980) and McCallum (1983).As a result of the rational expectations revolution the previously widely

used adaptive expectations mechanism was largely abandoned as it wasperceived to be a ‘‘backward-looking’’ mechanism. Furthermore, underadaptive expectations economic agents are allegedly making consistentforecast errors about which they do nothing. It should be pointed out thatin the early stages of the development of the jump-variable techniques forsolving rational expectations models some concerns were expressed aboutthe lack of any theory to explain the jump in economic variables as well asthe arbitrariness in the selection of the jump-variables in larger-scalemodels. Some of these issues were articulated by Burmeister (1980).�In spite of any of the foregoing lingering doubts, the rational expecta-

tions viewpoint has come to dominate in macroeconomic modeling. Ex-pectational schemes such as adaptive expectations or variants of it came tobe perceived as somehow ‘‘irrational.’’In our critical reappraisal of the jump-variable techniques we shall argue

that, if the economic variables are allowed to follow the unstable paths thatexist under both adaptive expectations and rational expectations, theneconomic forces will eventually come into play that stabilize the motion ofthe economic variables towards an attractor on which they exhibit self-sustaining endogenous fluctuations. The exact nature of the attractordepends on the dimension of the model under consideration. When oneallows expectations to tend to the perfect foresight (rational expectations)limit in this framework, we see that it is the expectational variable whichjumps discontinuously. However in our framework this jump is imposedby the system dynamics rather than being arbitrarily imposed by theeconomic model builder.We expound the discussion of the standard approach to the jump-

variable technique within the framework of the Cagan model of monetarydynamics. We choose a version which allows for lagged portfolio adjust-ment in the money market. We do this in order to make more explicit thesaddlepoint structure of the dynamics. The classical discussion of thismodel (see, e.g., Turnovsky 1995) assumes instantaneous portfolio adjust-ment, in which case the saddlepoint structure becomes degenerate in acertain sense. We should also point out, however, that there are goodtheoretical and empirical reasons for assuming lagged portfolio adjustment(see, e.g., Tsiang 1982, Kearney and MacDonald 1985).

� In this connection it is also worth recalling the comment of Blanchard (1981, p.135) who, inapplying the jump-variable technique to the value of the stockmarket, states: ‘‘Following astandard if not entirely convincing practice, I shall assume that q always adjusts so as toleave the economy on the stable path to equilibrium.’’


Our version of the Cagan monetary dynamics model is expressed as

m� p� f (�), (1.35)

m��m, (1.36)

p� ��(m��m)��

�(m� p� f (�)), (1.37)

�� (p� � �). (1.38)

Here m� lnM (M�nominal stock of money), p� lnP (P� price level),��E(p� )� expected rate of price inflation, and �

�,�� are speeds of adjust-

ment of price and inflationary expectations respectively.The money demand function f is assumed to satisfy f � � 0. For the

moment we simply assume

f (�)�� a�, (a� 0), (1.39)

which is the standard assumption in the traditional jump-variable litera-ture. However, we shall have more to say about the specification of themoney demand function later.We have deliberately specified the expectations mechanism (1.38) as

adaptive. However, by setting the speed of adjustment �� we canrecover the perfect foresight limit

�� p� . (1.40)

Under adaptive expectations the system (1.35)—(1.38) reduces to the two-dimensional dynamical system

p� ��(m� p� a�), (1.41)

�� (��(m� p)� (��a� 1)�). (1.42)

Under perfect foresight (�� ), (1.41) and (1.42) reduce to the one-dimensional system

p� �1

(a� 1/��)(p�m). (1.43)

Note first of all the equilibrium values

p� �m, �� 0 (1.44)

under both expectation schemes.In the following analysis we consider only the case

a� 1/�� 0, (1.45)

on the argument that the speed of price adjustment is quite high. We note


in this regard that the traditional discussion (e.g. Turnovsky 1995) sets��.The eigenvalue structure of the adaptive expectations system (1.41)—

(1.42) is easily calculated to be

��, ��

��

��2 �1��1�

4

�� , (1.46)

where�� a� 1/�

�� 1/��.

Since we are interested in this section in the limit ��, we assume �� issufficiently large that � � 0. We also assume that �� is sufficiently largethat �

�, �

�are real. Thus, for �� sufficiently large, both eigenvalues are

positive and the equilibrium is an unstable focus. This behavior is alsoillustrated in the phase diagram figure 1.1.Note that as �� the eigenvalues behave like

��, ��

��1

�,��

1

�� . (1.47)

Thus, as the perfect foresight limit is approached, one eigenvalue remainspositive but finite, whilst the second becomes an infinitely large positivenumber. This behavior is indicated in figure 1.1 by the vector field withdouble arrowheads. In the perfect foresight limit all trajectories moveinfinitely quickly in the � direction to the left or right depending on therelation of the initial value to the line �� 0.Now consider the one-dimensional perfect foresight system (1.43). Its

eigenvalue is given by

��1

a� 1/��

� 0. (1.48)

We note from (1.47) that

�� (��),

consistent with the view that perfect foresight is the limiting case ofadaptive expectations.Clearly the dynamic behavior under perfect foresight is unstable. The

dynamics are illustrated in figure 1.2.We see that all trajectories except the one satisfying p(0)�m are un-

stable. It is assumed that previously the economy was settled at theequilibrium p� and that at t� 0, money supply has increased to m to createthe new equilibrium p� �m. The jump-variable technique asserts that econ-omic forces will come into play such that the price level will jump discon-


Figure 1.1 Phase diagram of the dynamics under adaptive expectations

Figure 1.2 Instability in the perfect foresight limit

tinuously from p��to m at t� 0 so as to place the economy on the stable

manifold

p(t)�m,∀t.

This solution to the instability problem is conceptualized (justified) by the


procedure of integrating equation (1.43) forward over the interval(t,T )(T� t). Thus

p(t)� [p(T )e� ��]e!��1

� �

!

e�!��m(s)ds

� [me� �� (p(0)�m)]e!��1

� �

!

e�!��m(s)ds. (1.49)

Letting T��

p(t)� [p(0)�m]e!��1

� ��

!

e�!��m(s)ds. (1.50)

Sargent andWallace (1973) argue that p(t) must remain bounded as t��.It is clear that this can only be the case if

p(0)�m,

which again corresponds to the price jump p��m in figure 1.2. Now (1.50)

reads

p(t)�1

� ��

!

e�!��m(s)ds, (1.51)

which is the so-called ‘‘forward looking’’ solution of the rational expecta-tions approach. Through it, knowledge of future changes in m can beimpounded into prices at time t. This forward-looking or anticipatingbehavior is claimed as one of the great advantages of the rational expecta-tions approach. However at this point we want to stress that all of this iscrucially dependent on the market/economic agents in this economy beingable to compute precisely and instantaneously the required jump p�

��m at

t� 0. This in turn requires economic agents to have, or behave as if theyhave, complete knowledge of the model of the economy that they inhabit.For this reason we would argue that the ‘‘forward-looking’’ solution inrelation to equation (1.58) should more appropriately be termed the ‘‘fullknowledge of the model’’ solution.Some of the weaknesses of the jump-variable technique were well recog-

nized by its early proponents. First of all it requires on the part of economicagents complete knowledge of the model, instantaneous and low costaccess to market data, and a high degree of (low cost) computationalpower. These are required if the agents are to be able to compute � � p� ateach t (this assumes no information lags) and to compute precisely thejump p�

��m. This latter calculation becomesmore demanding the larger is

the model. Furthermore, if the model contains some nonlinearity the


Figure 1.3 Jump to linearization of stable manifold

calculation of the stable manifold becomes a further nontrivial computa-tional problem faced by the economic agents. It will not suffice to jumponto the linearization of the stable manifold since trajectories starting on itwill be unstable as shown in figure 1.3.Secondly, there is no theory to explain the jump p�

��m. Somehow the

marketmakes it happen. If the jump occurs at time t�then themodel under

discussion applies for t� t�and for t� t

�, but at the precise point of major

economic interest, t� t�, the model is ‘‘switched off ’’ and the jump, for

which we have no model, occurs.The foregoing criticisms were acknowledged by the original proponents

of the rational expectations (jump-variable) paradigm. They generallyseemed (implicitly) to argue that the approach was nevertheless worthpursuing as it provided a polar case, with the implication that morerealisticmodels reflecting the informational and computational limitationsof real world economic agents would not behave very differently. However,in order for the polar case to be useful, the modeling frameworkmust enjoya certain ‘‘robustness’’ property, namely, that it must have the same quali-tative properties as models which are ‘‘close’’ to it in some topologicalsense.An important, but unfortunately ignored, criticism of the jump-variable

technique has been made by George and Oxley (1985) and Oxley andGeorge (1994). They make the point that the jump-variable techniqueyields a structurally unstable model. This implies that the polar case doesnot enjoy the robustness property referred to earlier. To some extent this isa restatement of the criticism that agents need complete knowledge of themodel (and its coefficients). Thus, if (in matrix notation) the real model is


Figure 1.4 The true and the perceived system

x� �Ax, whereas agents due to imprecise knowledge of the coefficientsperceive x� �A� x, where �A�A� � �� (� some small quantity), then theywill get the jump wrong and proceed on some unstable path as shown infigure 1.4; i.e., they jump from E

�to N� instead of E

�to N. Under the

dynamics of A (the true system) the initial point N� is on an unstabletrajectory.An aspect of the jump-variable technique that has not been discussed is

the very rationale for its use. This rationale seems to be that, unless it isapplied, the economic variables will move off to �� or �� along theunstable paths of the saddlepoint. However this rationale seems to ignoretotally the possibility (and indeed strong likelihood) that other economicpressures will come into play as economic quantities move further andfurther from their equilibrium values. Typically, the models to which thejump-variable technique is applied have in the background some storyabout the allocation of wealth between alternative assets. Certainly thiswill be the case for the models we consider in this book. Furthermore, theexpectational variable usually plays a role in determining the return differ-ential between the alternative assets. Thus, in the Caganmonetary dynami-cs model, the expected rate of inflation plays a role in determining theallocation of agents’ wealth between money and physical capital. As theexpectational variable moves far from equilibrium the allocation of wealthwill move to an extreme, i.e., all in money or all in physical capital.


Figure 1.5 Nonlinearity in the money demand function

Thus the asset demand function must be bounded above and below asillustrated in figure 1.5 for themoney demand function of the Caganmodel.As �� money loses all value as a store of wealth and agents prefer tohold wealth in the form of physical capital. Hence money demand sinks tosome minimum level f

�determined say by transactions demand. At the

other extreme as �� the expected rate of price deflation is so greatthat agents want to hold a minimum amount of wealth in physical capitaland the maximum amount possible in money. The upper limit f

"thus

represents the largest fraction of wealth that it is possible to hold in theform of money.�Thus we see that it is essential to model the asset demand functions as

nonlinear functions. The linear (or linearized) asset demand functions ofthese models (e.g., equation (1.39) above) only hold close to equilibriumand fail completely in capturing these important portfolio adjustmenteffects. It is important to stress that these nonlinear effects do not result insome second-order effect to the dynamical picture obtained from thelinearized model. Rather, they are crucial to the dynamical behavior of themodel and result in a qualitatively different dynamical picture. It is herethat the significance of the point about structural instability made byGeorge and Oxley is appreciated.In our review of the jump-variable technique we showed how perfect

foresight can be viewed as the limiting case of adaptive expectations as thespeed of adjustment of expectations, ��, tends to�. The eigenvalue analy-sis of adaptive expectations with �� sufficiently large indicated two positive� It is also possible to arrive at such a nonlinear money demand function by using a

two-period utility maximizing model, see Chiarella (1990, ch. 7).


eigenvalues. As ��, one eigenvalue tended to the one of the perfectforesight model whilst the second tended to�. We see from figure 1.1 thatthis infinite eigenvalue represents infinitely fast motion in the � direction,whereas motion in the p direction remains finite. This picture is in sharpcontrast with that from the jump-variable technique which imposes aninfinitely fast movement on p at t� 0. The infinitely fast eigenvalue ishidden from sight in any analysis which starts from the perfect foresightmodel (i.e., equation (1.43)). The fact that the one-dimensional system (1.43)is the limiting case of the two-dimensional system (1.41)—(1.42) has not beenappreciated in the literature. There is, in fact, a mathematical theory ofsuch dynamical systems which ‘‘lose’’ a dimension as the coefficient on aderivative term tends to zero (i.e., as 1/�� 0 in equation (1.42)). These areknown as singularly perturbed systems, and an excellent account for ourpurposes is given by Andronov, Vitt, and Chaikin (1966, chapter 10). Theimportant point that emerges from this theory is that, to obtain the truedynamic picture of the lower dimensional system (i.e. (1.43)), it is necessaryto consider the higher dimensional system (i.e., (1.41)—(1.42)) and proceed tothe limit (i.e., ��).Analysis of the dynamical system (1.37)—(1.38), when the money demand

function has the qualitative shape in figure 1.5, yields the phase diagramshown in figure 1.6. Chiarella (1986, see also Flaschel and Sethi 1999), hasshown that, for �� sufficiently large, trajectories are attracted to the stablelimit cycle. Furthermore, in the perfect foresight limit as �� the limitcycle tends to a limiting limit cycle or relaxation cycle as shown in figure1.6. On this perfect foresight relaxation cycle prices move continuously, butnonsmoothly, and expectations move discontinuously, as shown in figure1.7, which illustrates motion from the initial point. A more completediscussion of the economics behind figures 1.6 and 1.7 is given in Flaschel(1993).

1.6.2 The empirical evidence

The empirical literature on expectations formation that we review is drawnfrom studies of foreign exchange markets. This choice reflects the wideavailability of data for such markets. However, our choice is also guided bythe consideration, that if there is any market in which the rational expecta-tions view should hold, it is the foreign exchange market. This market ishighly liquid, not dominated by any one, or group of, economic agents, andinformation in it and about it is disseminated very rapidly indeed. We firstdiscuss papers which study survey data of exchange rate expectations andthen an econometric study based on a structural open economy macro-economic model.


Figure1.6

Relaxationoscillationininflationaryexpectations

Figure 1.7 Time series presentation of the relaxation oscillation

Cagan (1991) analyses weekly data on forward mark—pound exchangerates for 1921—1923. He finds that predicted changes in the spot rate basedon these forward rates are biased downward substantially and do not passthe standard tests of rationality.� He further finds that adaptive expecta-tions outperform rational expectations during this period. Cagan suggeststhat this could be explained by a gradual adaptation of the market to thenew volatile regime of German hyperinflation. Cagan’s comment that, ‘‘ifthe coefficient� were to become large enough, the adaptive formula ap-proaches the strict definition of rational expectations’’ is quite consistentwith the modeling viewpoint we are proposing, namely, to analyze rationalexpectations (perfect foresight) as the limit of adaptive expectations as thespeed of adjustment goes to infinity. The studies of Frankel and Froot(1987, 1990) use various survey data of exchange rate expectations andperform various tests of the rationality of expectations formation. Theyfind evidence of systematic expectational errors and the operation of somekind of adaptive scheme. They also conclude that heterogeneous expecta-tions play an important role in determining market dynamics. In particu-lar, they suggest that the simultaneous existence of fundamentalist (i.e.,adjusting expectations to a long-run equilibrium) and chartist (i.e., adjust-ing expectations to recent price trends) elements may more realisticallycharacterize the operation of foreign exchange markets. Allen and Taylor(1990) come to a similar conclusion.Motivated by the study of Frankel and Froot, Papell (1992) estimates a

stylized version of the Dornbusch (1976) model of exchange rate dynamics.

� I.e., that under rational expectations, prediction errors are unbiased and serially uncor-related for successive data points. � I.e. coefficient of adaptive expectations.


He formulates the model under both rational expectations and adaptiveexpectations. The model is estimated for seven countries and he is unableto distinguish econometrically between adaptive expectations and rationalexpectations. This econometric evidence is also consistent with our view ofcasting the modeling of expectations within an adaptive expectationsframework and treating rational expectations/perfect foresight as a limit-ing case. Papell’s empirical results would be consistent with the estimatedvalue of �� being large but less than infinite.

1.6.3 Heterogeneous expectations

In view of the empirical evidence of Frankel and Froot and Allen andTaylor discussed in the last subsection, we seek to model expectations as aweighted average of fundamentalist and chartist elements. We do so inalmost the simplest manner possible, but do indicate how this approachcould be elaborated upon considerably.We develop our heterogeneous approach to expectations modeling

within the context of the Cagan monetary dynamics model. However, wewill apply the same general approach to other expectations schemes used inthis book.We use subscript 1 to denote the group of economic agents who form

expectations using chartist (i.e., technical analysis or time series) techniquesand subscript 2 to denote the second group of economic agents who formexpectations using fundamentalist (i.e., theory/model based) techniques.Considering first the chartists, the simplest type of rule they could follow

is described by the adaptive expectations scheme

�� (p��). (1.52)

On the other hand, the fundamentalists would form expectations in aforward-looking way according to

�� (p*��), (1.53)

where p* is the fundamentalists’ view on the long-run or equilibrium rate ofinflation. This could be based on some model of the economy or theso-called p-star inflation rate theory.�

A direct comparison with Papell is difficult as his expectation variable is on the level ofexchange rates rather than on the change in exchange rates as in Dornbusch’s and our ownapproaches.

A farmore general class of ruleswouldbe �(t)� �!��

�(t� s)p(s)ds, where� is, for example,some kind of exponential weighting function.

� The p-star inflation rate theory as it is used by the FED or the German Bundesbank isbasedon the quantity theory ofmoney and the concept of potential output, which is used toestimate the p-star price level by means of this theory (see chapter 6 for details).


We assume the proportions of chartists and fundamentalists to be fixedat � and (1� �).�Thus the average expected rate of inflation, �, is given by

�� (1� �)�

�, (0� �� 1). (1.54)

From (1.52) and (1.53), and performing some manipulations, we find that

�� (p��)� (�� )(1� �)(p*��), (1.55)

where p�� p� (1� �)p*.We note that, in order to obtain an aggregated expectations mechanism

that only depends on aggregate expectations, we need to set

�� , (1.56)

in which case (1.55) reduces to

�� (p� �)� (1� �)�� (p*� �). (1.57)

By setting �� and (1� �)�� we obtain the aggregate expec-tations mechanism

�� (p��)��(p*��). (1.58)

This mechanism, and possible extensions of it, will be employed through-out the book. It has the analytical advantage that it allows us to ignore thelaws of motion of individual expectations �

�and �

�.

We note in passing that the proportion of chartists is given by

��

�� in this particular form of our aggregate expectations mechanism. A mech-anism of this type has been employed by Groth (1988) in his study of theconsequences of a combined adaptive and forward looking expectationsscheme.For most of our dynamic models in this book we shall simply take p* to

be ��, the steady state equilibrium level of �, so that (1.58) specializes to

�� (p��)��(��). (1.59)

In chapter 4, however, we shall use the full p-star concept as used by theFED of the United States of America or the German Bundesbank in orderto represent the forward-looking component in the average expectationsformation.Equation (1.59) nests many special cases which are considered in the

� A more complete and satisfactory treatment would allow their proportions to evolveaccording to the success of the two predictors in the recent or more distant past. We returnto this point later.


literature. Thus, �� 0 (i.e., 0��, �� 1 so that chartists dominate)yields adaptive expectations. The case �� 0, 0�� (i.e., � � 0 so thatfundamentalists dominate) yields modified regressive expectations. Thecase ��,�� (i.e., �� , � � 1, so that chartists dominate andhave infinitely fast speed of adjustment) corresponds to myopic perfectforesight. Finally the case �� ,�� (i.e., �� , � � 0, so thatfundamentalists dominate and have infinitely fast speed of adjustment)corresponds to asymptotically rational expectations (see Stein 1982).As we have mentioned earlier, a more complete treatment of expecta-

tions would allow the fraction � of chartists to evolve according to thedynamic evolution of the key economic quantities. Approaches to such amore complete treatment have been explored by both Brock and Hommes(1997) and by Sethi (1996). The task of integrating such evolutionaryexpectations mechanisms into the models discussed in this book remainsan important topic for future research. A preliminary analysis has beenundertaken by Chiarella and Khomin (1999).

1.7 A new integrated approach to Keynesian monetary growth

Now that we are near the close of this chapter, let us briefly summarizewhat we have claimed will be the achievements of this book on Keynesianmonetary growth dynamics.

1.7.1 The basic Keynesian prototype model: IS—LM growth dynamics

Progress in our modeling of this type of growth analysis takes its point ofdeparture from Sargent’s (1987, part I) framework for macroeconomicmodel building with its three sectors (households, firms, and the govern-ment) and its five markets (labor, goods, money, bonds,� and equities�)and also from our generalized growth models of Tobin and Keynes—Wicksell variety. These model types are also investigated, although from adifferent point of view, in Sargent (1987, part I), the latter type in the limitsituation of perfectly flexible prices.The AS—AD growth model is generally considered as allowing for two

important variants (here discussed in section 1.5) namely the classicalvariant with perfectly flexible prices and wages and the Keynesian variantwith only sluggish changes in the money wage due to labor market imbal-ances and inflationary expectations. The proper Keynesian variant of this

We use the term modified regressive expectations, as in the language of Dornbusch (1976)regressive expectations would read �� (��

� p).� As the instrument of finance of the government.� As the instrument of finance of the firm.


approach, however, should, as we have seen, integrate capacity utilizationproblems of firms and exhibit sluggishness not only with respect to wages,but also with respect to prices. This model type will considered in detail inchapter 4.A simple change and extension of the AS—AD growth model type seems

therefore to be sufficient in order to arrive at a proper and internallyconsistentKeynesianmodel, now of IS—LMgrowthwith prices respondingsluggishly to imbalances in the market for goods (measured by the rate ofcapacity utilization within firms) and expectations on wage inflation. Yetthis stage of disequilibrium growth theory demands one further change inthe setup of the model, since the varying degree of capacity utilization offirms should now also be taken into account in their investment behavior.From a formal perspective, these are small, but nevertheless unavoid-

able, changes in the overall structure of AS—AD growth models typicallyused for analyzing the growth of monetary economies from a Keynesianperspective. Yet, as chapter 4 will demonstrate, the dynamical analysis ofprocesses of monetary growth is considerably changed thereby, particular-ly in comparison to the predecessor model of Keynes—Wicksell type ofchapter 3. There is thus a significant qualitative change involved whengoing from the Keynes—Wicksell (or the limiting AS—AD) growth model tothe IS—LM growth model, with its typical wage—price spiral of demand-pull and cost-push type. This gives the reason why we call this IS—LMgrowth model the fundamental Keynesian prototype model of monetarygrowth, inflation, and fluctuations in the utilization rates of both capitaland labor, while its predecessor models of Keynes—Wicksell or AS—ADtype are considered as still incomplete from a Keynesian perspective. Weshall furthermore show in chapter 5 (and chapter 4) that it is not difficult toincorporate into this Keynesian prototype model smooth factor substitu-tion (and technological change, more elaborate taxation schemes, morecomplex wage—price spirals) in order to make it at least as general as theAS—AD growth model that preceded it.

1.7.2 The Keynesian working model: Keynes—Metzler monetary growthdynamics

There are, however, some features of this basic as well as generalKeynesianprototype model observed in chapter 4 which suggest that there are stillsome fundamental problems present in this model type. Using the dynamicmultiplier process as a further justification of the temporary IS—LM equi-librium position indicates stability problems for this equilibrium positionfor certain sets of parameter values. Furthermore, there are three possiblescenarios for these parameter values with quite different comparative static


properties of the IS—LM equilibrium which (to some extent) are separatedfrom each other through discontinuities in the model’s behavior. Thissuggests that ways have to be found by which the partial instability of thedynamic multiplier process can be integrated and overcome without lead-ing to particular situations where the model is ill defined.Chapter 6, the core chapter of the book, which finally presents the

fundamental working model of our Keynesian monetary growth analysis,integrates and thereby removes the observed multiplier instabilities (work-ing in the background of the model of chapter 4) by just taking note of onesimple implausibility left in the model of this chapter, namely its asymmet-ric treatment of the adjustment speeds of wages and prices on the one hand(which are finite), and that of quantities on the other hand (which — due tothe assumed goods market equilibrium — is infinite). There is, indeed, awell-known (theoretically as well as empirically relevant) way out of thisproblematic asymmetry, namely, the appropriate integration of the Metz-lerian inventory adjustment mechanism into the prototype model of chap-ter 4. Allowing in the spirit of this inventory model for disappointed salesexpectations one has then to formulate how these expectations and theoutput decisions of firms are in fact revised (on the basis of observed sales),and this in the light of their accumulated factual inventories compared totheir desired inventory level in the framework of a growing economy.This task is solved in chapter 6 along lines proposed by Franke and Lux

(1993) and Franke (1996), leading to a model that is on the one hand fairlyelaborate in its price, wage, quantity, and expectations adjustment mech-anisms (implying at least six laws of motion for its central state variables),but which on the other hand is still of a very traditional type as far as itsbasic modules (when considered in isolation) are concerned. We shallpresent in chapter 6 an analysis of this model of monetary growth from theperspective of appropriately isolated typical subdynamics of it, as well asfrom the perspective of their integrated interaction, and shall find that theisolated perspectives, often well known from the literature on Keynesiandynamics, do not properly inform us on their behavior when they areintegrated and interacting with each other. This implies that there is anurgent need for a further understanding of this integrated Keynesianmonetary growth dynamics, which can be satisfied in the present book onlyin a preliminary way.Chapter 6 therefore supplies on the one hand an internally consistent

Keynesian model of monetary growth of fairly traditional type and with awell-balanced structure, but it also shows on the other hand that we stillhave a very limited understanding of the working of its dynamics due to thehigh dimensionality involved. We have therefore arrived at a benchmarkmodel, whose further detailed analysis is urgently needed in order to really


understand the achievements of the generally low-dimensional analyses ofeconomic dynamics of the more recent literature on (Keynesian) monetarygrowth.The Papers and Proceedings of the American Economic Review have

recently published a discussion on ‘‘Is there a core to practical macro-economics that we should all believe?’’ with contributions by Blanchard(1997), Blinder (1997), Eichenbaum (1997), Solow (1997), and Taylor (1997).Our reading of this discussion is that the working model of Keynes—Metzler type we will arrive at in chapter 6 (and which we extend further onthe supply side in chapter 7) provides a general prototype of an integrated(traditional type of) macrodynamics of the short, the medium, and the longrun that is in many respects closely related to this discussion, also withrespect to the term ‘‘practical.’’ Our working model can at the least be usedas a point of departure for the further discussion of integrated macro-dynamics with demand- and supply-side features and their relationshipswith structural macroeconometric model building. In this regard we alsorefer the reader to the discussion on macroeconomic modeling in a chang-ing world in Allen and Hall (1997).

1.7.3 The path ahead

We start the analysis of the dynamics of Keynesian monetary growth inchapter 3 with the generalized Keynes—Wicksell approach, which employsexternal disequilibrium concepts both on the market for labor as well as onthe market for goods. Compared to this starting point, chapter 6 may beconsidered as the truly Keynesian completion of the project begun withthese Keynes—Wicksell prototype models, since it returns to assumingdisequilibrium in both external labor and external goods markets (besidesallowing for endogenously determined disequilibrium within firms asmeasured through the rate of capacity utilization). Wemay therefore claimthat the proper integrated model of Keynesian monetary growth, as thetraditional starting and reference point for all further models of this type,has thereby been found (and is not at all easy to analyze).Yet, the analysis of our model cannot finish at this point, though it will

not be extended very much further within the scope of this book. In a finalstep, in chapter 7, we add to our working model with sluggish price as wellas quantity adjustments an endogenous determination of natural rates ofemployment and growth, as well as a sluggish adjustment of the rate ofemployment in the light of over- or undertime work within the firm, andshow that some new and interesting dynamical features will be establishedthereby. At least equally important, however, would have been to allow fora more elaborate structure of financial markets, or for more developed


mechanisms of wage and price settings, and further important extensionsor refinements of the modules of the working model listed in the finalsections of chapter 7. These are, however, topics that must be left for futureinvestigation.

1.7.4 One final methodological remark

Throughout this book we rely on models of monetary growth which arebased on linear behavioral or technological relationships for as long as thisis possible and meaningful.�� This allows us to concentrate on and toinvestigate the existence and implications of unavoidable ‘‘natural’’ non-linearities first. These models generally allow an explicit calculation oftemporary equilibrium positions and can often be adequately studied as totheir dynamic behavior by means of eigenvalue calculations and the asso-ciatedHopf bifurcations. Specific further nonlinearities (in money demand,in investment demand, in production, etc.) should and will be introduced ata later stage and justified to some extent. They are in particular needed ifglobally explosive dynamics are implied through the linear version ofmonetary growth (with only ‘‘naturally’’ occurring nonlinearities), whereone is of course compelled to add and to discuss the forces that may or willcome about when the economy departs by too much from its steady-stateposition.Despite these occurrences, the book can nevertheless be based in many

of its parts on linear behavioral and technological relationships (formethodological reasons and for reasons of simplicity), and it shows in thisway that there is much to do and to discuss even on this preliminary level ofthe analysis of the dynamics of monetary growth, where only unavoidablenaturally occurring nonlinearities are taken into account.

1.8 Mathematical tools

We may say that recently, the non-market-clearing as well as the market-clearing approaches to macroeconomics apply similar technical tools instudyingmacrodynamics. Both borrow from recent advances in themathe-matical literature on nonlinear discrete or continuous time dynamicalsystems. In general the dynamic analyses that exist in the literature areconfined to dimensions two or three (even in continuous time) and thus

�� In the case of money demand we specialize further and make use of Taylor expansions offunctions with, on the one hand, output and, on the other hand, nominal interest andwealth as arguments and use in addition to that the capital stock K as a proxy for realwealth; see Chiarella et al. (1998) for a full treatment of wealth effects on the market formoney as well as for goods.


much less ambitious than the continuous time dynamical systems that weare analyzing in this book as we approach the workingmodel of Keynesianmonetary growth and its extensions in chapters 6 and 7.We want to mention the particular technical tools that are involved in

our study. We use the established toolbox of local stability analysis basedon linear approximations and of the global stability analysis of planarsystems.The local tools are basically given by the necessary and sufficient condi-

tions of the Routh—Hurwitz theorem (see Gantmacher 1959, ch. 15, andBrock andMalliaris 1989, ch. 3, in particular), and they are here applied indimensions two and three (see also the appendix to this section). Asmentioned, we do, however, also investigate continuous time dynamicalsystems of much higher dimension. There, we can prove local asymptoticstability by approaching them from lower dimensional models where wealways check at each step that the determinant has the appropriate sign forlocal asymptotic stability and make use of the fact that eigenvalues dependcontinuously on the parameters of the dynamics.For planar systems we prove global stability (boundedness of the dy-

namics in appropriately chosen economic domains of the considered phasespace) by making use of invariant sets to which the Poincare—Bendixsontheorem can then be applied to prove the existence of periodic motions(generally attracting limit cycles), or of (local or global) Liapunov functionswhich have the shape of a sink and which can also be used to characterizeinvariant subsets of the phase space (where often the trajectories pointinwards and thus converge to the steady state). The Poincare—Bendixsontheorem is discussed in detail in Hirsch and Smale (1974, ch. 11) andArrowsmith and Place (1990, ch. 3).�� Both of these books also present thetheory of Liapunov functions in their chapters 9 (section 3) and 5 (section4), respectively (see also Brock and Malliaris 1989, ch. 4).We also make use of special limit cycle configurations, so-called relax-

ation oscillations (or limit limit cycles), which arise when certain speeds ofadjustment parameters approach infinity (see Arrowsmith and Place 1990,4.4, Chiarella 1990, 2.6, and Strogatz 1994, 7.5).Also, local Hopf bifurcation theory is frequently used in this book.

Presentations of this material that are not too technical are provided inGuckenheimer and Holmes (1983, ch. 3), Arrowsmith and Place (1990, 5.5),Wiggins (1990, 3.1), Perko (1993), and Strogatz (1994, ch. 8).Numerical methods and problems are discussed in Parker and Chua

(1989) and by Hairer, Nørsett, and Wanner (1987), in particular in theformer, with special emphasis on the treatment of chaotic systems or

�� See also Wiggins (1990) and Perko (1993).


strange attractors (defined by excluding points of rest, limit cycles andquasi-periodic motion from discussion). Our calculation of bifurcationdiagrams and Liapunov exponents of high dimensional dynamical systems(where nonlinearities needed to generate complex dynamics may be con-siderably weaker than in low dimensional systems) follows the proposalsmade by Parker and Chua in this respect. Strange attractors and chaoticmotions are discussed in Guckenheimer and Holmes (1983), Wiggins(1990), and Strogatz (1994).There exist meanwhile also a variety of detailed surveys on dynamical

tools for economists where traditional and newer methods for analyzingnonlinear dynamics are presented and exemplified by dynamic economicmodels in particular. See in this regard Medio (1991, 1992), Tu (1994),Gandolfo (1997), Lorenz (1997), and Puu (1997), where the techniquesdiscussed above are also presented and illustrated by means of examplesfrom the economic literature.All these sources fully cover the dynamical tools that we will use in this

book and should be consulted by the reader for the details of the mathe-matical propositions we derive as we proceed from chapter 2 to the higherdimensional structures that we finally obtain for the working model (andits extensions) in chapters 6 and 7.Concerning programming tools used in the numerical investigations of

this book we have primarily relied on the programming languageGauss byAptech Systems and on SND, a Windows 95 package of our own forsimulating continuous and discrete time dynamical models, developed byA. Khomin (see Chiarella, Flaschel, and Khomin 1998 for further details).Occasionally we have used Locbif (see Khibnik et al. 1993) to drawbifurcation curves in two parameter spaces, and DMC by G. Gallo, (seeMedio 1992), for the presentation of limit cycles and the like.

Appendix

We gather here for easy reference the details of the Routh—Hurwitz condi-tions for the stability of the three-dimensional dynamical system

d

dt�x�x�x��J�

x�x�x� , (1.60)

where


J��a��

a��

a�

a��

a��

a�

a�

a�

a� . (1.61)

We recall first that the principal minors J#(the determinants obtained by

eliminating the row and column containing a##) are given by

J��

a��

a�

a�

a � , J

��

a��

a�

a�

a� , J

� �

a��

a��

a��

a�� .

The characteristic equation of the dynamical system (1.60) turns out to be

�� trace (J)�� (J��J

��J

)��det J� 0. (1.62)

The Routh—Hurwitz necessary and sufficient condition for the stability ofthe system (1.60) can be expressed as

trace (J)� 0,J��J

��J

� 0,

det J� 0,

and

�trace (J)(J��J

��J

)� �J �� 0.


2 Tobinian monetary growth: the(neo)Classical point of departure

In this chapter we introduce a Classical reformulation of the Tobin (1965)model of monetary growth, with fixed proportions in production andClassical saving habits. Section 2.1 presents the standard full equilibriumversion of this model type, while section 2.2 considers the money-marketdisequilibriumextension of it which was extensively studied in the seventiesand the early eighties. By means of these reformulations of models of theliterature we shall recapitulate some of the important results obtained forthis monetary growth model type concerning non-superneutrality of thesteady state and the pure Tobin effect, instability of the steady state due towhat we call the Cagan effect in the money demand function, and certainnew limit cycle results which can be built on such local instability.The chapter then proceeds, in section 2.3, by providing some new

extensions to these Tobin type models. First, since the Tobin models relyon an elaborate form of Say’s Law for the market for goods, it is very easyand natural to extend the model to labor market disequilibrium. Thisdisequilibrium is here due solely to capital shortage or abundance and iscompletely decoupled from the situation on the market for goods. Thisextension, which basically reformulates the wage—price sector of the model,generally increases the dynamic dimension of the model by two to four,since there is now room for disequilibrium fluctuations of the real wage aswell as fluctuations in the growth rate of the capital stock. An importantquestion in this context will be how cycle mechanisms known to exist forthese subdynamics (which are related to the names of Goodwin and Rose)combine with the above Cagan cycle on the nominal side of the model andgenerate therebymore complex dynamics in comparison to such limit cyclesubcases. Due to the difficulties in analyzing such nonlinear continuoustime dynamics of dimension four, only some general results and a numeri-cal investigation of the four-dimensional case will be possible here.Section 2.4 then presents another generalization of this approach to

monetary growth by extending it to include bonds and government debt

69

and by falling back on the situation of general equilibrium for the fourmarkets of this extended model.� We here investigate the consequences ofalternative concepts of disposable income that have been used in theliterature in models which embrace the consequences of government debt.The dynamics of the model will then again be of dimension two, and willbecome identical to the initial case (section 2.2) of no government debt ifBarro’s concept of disposable income is employed.�Finally, in section 2.5 we synthesize the previous two extensions of the

Tobin type models into a general neoclassical model of monetary growthwith disequilibrium onmoney and labor markets. The intention here is notsomuch to provide a really general version of this model type, but rather toformulate a complete neoclassical model of monetary disequilibriumgrowth which is as consistent as possible, in particular with respect to thebudget constraints of the various sectors of the economy. This modeldemonstrates that there are two paths for future developments. Either areturn to its full equilibrium version as a research paradigm, which is thegeneral viewpoint of most of the models of monetary growth of the recentpast, or to develop the general disequilibrium version further by offering inparticular more convincing descriptions of the particular disequilibriumsituation that is responsible for price increases in the Tobin model. It is thislatter viewpoint that is taken up in the next two chapters where a system-atic variation of the general Tobin model is performed that finally trans-forms it into a proper Keynesianmonetary growthmodel with problems ofeffective demand and demand pressure inflation descriptions. As an inter-mediate step, we shall, in the next chapter, confront the general Tobinmodel with a general Keynes—Wicksell model that is obtained from theliterature of the seventies by again completing the model structures of thatliterature in the way already indicated for the Tobin model.It is our hope that such systematic extensions and modifications of

prototype models of descriptive monetary growth theory will convince thereader that there still is a high potential, which is unexploited but valuable,for future developments in these approaches. This potential has simplybeen ignored during the progress that has been made in the theory ofmonetary growth in the recent past.

� There is not yet a market for equities in the Tobin models of monetary growth, due to thelack of an independent investment function in this type of model.

� See Barro (1974) and Sargent (1987, I.10).


2.1 The basic equilibrium version of Tobin’s model of monetarygrowth: superneutrality and stability?

We now start our investigation of the consequences of introducing moneyas an asset in a dynamic Classical framework in a Tobinian way which ishere made as complete and consistent as possible by the specification of thebudget constraints of all three sectors: households, firms, and the govern-ment.Note here in particular that the flow constraint of households is oftenrepresented in this type of literature by an identity of the kind K� �

(M� /p)� s�(Y� �K��L� (M� /p)), if the expected rate of inflation equals

the actual one. As the following shows, this is, however, not a good startingpoint for the presentation of the model, since this identity already is anaggregate of the budget restrictions of households and the government asthe following discussion should make clear.

The equations of the model are:

1 Definitions (remuneration and wealth):

��w/p, u��/x,� � (Y� �K� �L)/K, (2.1)

W�M/p�K. (2.2)

2 Households (workers and asset holders);

W�M/p�K,M� h�pY� h

�pK(r� � (�� )), r� � const.,

(2.3)

Y��

��K�M

p� �T, (2.4)

C��L� (1� s�)Y��, s�

� 0, (2.5)

S��L�Y�

��C�Y� �K�T�C

� s��K�

M

p��T��

M

p�, (2.6)

�M� /p�K� ,

L1 � n� const. (2.7)

3 Firms (production units solely):

Y� yK(�Y�� y�K),L�Y/x, y, x� const. (2.8)

4 Government (monetary and fiscal authority):

M� ��M,

�� const., (2.9)

See pp. xxii—xxiv for the employed notation.

71Tobinian monetary growth

G� g� K, g� � const., (2.10)

T�G�M� /p[S��T�G��M� /p]. (2.11)

5 Equilibrium conditions (asset markets):

M�M[K�K], (2.12)

M� �M� [K� �K� ]. (2.13)

6 Say’s Law on the market for goods for ‘‘full employment’’:

K� � S��S

��S�Y� �K�C�G�K� , (2.14)

L�L[K1 � n]. (2.15)

7 Combined adaptive and regressive expectations:

�� (p��)��(�� n� �),��#

� [0,�], i� 1, 2 (2.16)

The above model is subdivided into seven sections which will reappear inall later, more extended models. The formulations of the equations in thevarious sections of the model are not without repetitions, repetitions bywhichwe intend tomake the contents of and the connections between theseequations more transparent. This most basic of our models of monetarygrowth still abstracts from an independent investment behavior of firmsand can thus still ignore the market for equities. In the tradition of theTobin models of monetary growth of the seventies it also assumes thatgovernment expenditure is financed either by money or by taxes (govern-ment debt and bonds will be introduced at a later stage of modeling in thischapter).The equations (2.1) and (2.2) in the first section provide the definitions of

important macroeconomic magnitudes, namely, of the real wage �, thewage share u (x gross output per laborer), the actual rate of profit � and realwealthW, which here consists of real money balances and the capital stocksolely.We assume two groups of households in all of our models, workers and

asset holders (capitalists), for whom we are assuming Classical savinghabits, i.e., workers do not save (constant savings propensity s

�� 0) while

capitalists save a constant fraction (constant savings propensity s�) of their

perceived disposable income Y��. Since we assume in chapters 2—4 fixed

proportions in production (in order to simplify the exposition), and thusshall allow for (smooth) factor substitution only later on, we need such anassumption on differentiated savings habits in order to obtain a steady-state solution for such a model, which generally does not exist for fixedproportions technologies in the frequently assumed case s

�� s

�� s. This

latter assumption furthermore is further from reality than the one we


employ. Of course, assuming Kaldorian differentiated saving habits wouldbe even better (extended possibly also to other groups of savers such aspension funds for example). This extension is, however, completely avoidedin the present book and must be left for future investigations.As already indicated, the decision of workers is a trivial one here. They

just consume what they get as income in the current period (�L) andthereby fulfill their budget restriction in the simplest conceivable way. Thedecision of asset holders, by contrast, concerns, on the one hand, theallocation of their wealth between real balances (demand,M/p) and realcapital (demand, K) and, on the other hand, their savings decision (theaccumulation of new wealth).Due to the wealth constraint of asset holders (see equations (2.3)), we

only need to specify their stock demandM for money. The stock demandfor real capital is then obtained as a residual from this constraint. Themoney demand of capitalists is here postulated as is usual in Tobin modelsof monetary growth. That is, it depends on gross output Y as a proxy forthe volume of transactions and on the rate of return differential ��between capital holding andmoney holding, i.e., the rate of profit earned inproduction plus the expected rate of inflation (to be defined later on). Notethat we here employ a linearized function for describing this demand —which in general could be of the form M/(pK)�m(Y/K,� ��).In this book, we rely on models of monetary growth which are based on

linear behavioral or technological relationships for as long as this ispossible and meaningful in order first to investigate the existence andimplications of ‘‘natural’’ nonlinearities. These models generally allow anexplicit calculation of temporary equilibrium positions and can often beadequately studied as to their dynamic behavior from a local perspectiveby means of eigenvalue calculations and the like. Specific nonlinearities (inmoney demand, in factor relationships, etc.) should and will be introducedand justified at later stages to some extent. Nevertheless, the book is basedin many of its parts on linear behavioral and technological relationshipsand it shows in this way that there is much to do and to discuss even on thispreliminary level of the analysis of monetary growth dynamics where onlyunavoidable naturally occurring nonlinearities are taken into account.Equation (2.5) describes aggregate planned consumptionC as the sum of

the two components we have already described above by referring to �Land Y�

�as the perceived disposable income of workers and asset owners.

The latter perceived income is defined in a Hicksian way, i.e., it is based ondeductions that guarantee that real wealth, as defined in (2.2), stays intactshould all perceived income actually be consumed. In the present simple

See Chiarella et al. (1998) for the details of such an approach.


case this means that purchasing power losses of money balances must bereplaced through deductions� from actual disposable incomeY�� K�T of capitalists, i.e., from profit income after taxes T. For

reasons of simplicity with respect to workers’ behavior, we have assumedhere that all taxes are paid out of profits (just as all money is held solely bycapitalists).�Total private savings S

�(see equation (2.6)) is defined by total disposable

income of households minus consumption: �L�Y��C�

Y� �K�T�C and, of course, here is equal to the savings of capitalistsout of their perceived disposable income. The intended allocation of theirsavings is described by the flowmagnitudesM� /p,K� which in equilibrium,as shown below, give rise to subsequent changes in real balances m�M/p(m� �M� /p� p(M/p)) and real capital formation K� (d� demand). Finally, it is assumed in (2.7) that normal labor supply L grows with a

given rate n which is also equal to the growth rate of the total population.Later in this chapter and in other chapters we shall allow for overtimeworkwhichmeans that the numberL should not be interpreted as the maximumsupply of labor hours available in the economy in the models that follow.Section 3 of the model, (2.8), describes the behavior of firms in the most

elementary way available in a two-factor world. Firms produce outputwith a fixed proportions (linear) technology using capital and labor inputs,the former at full capacity. That is, they face no demand constraint on themarket for goods and no supply constraint on the market for labor, due tothe prevalence of Say’s Law on the market for goods (see (2.14)) and sincethere is full employment on the market for labor by assumption (asexpressed in (2.15)). Note here that labor demand L is determined by thelevel of the capital stock in each period in the following simple wayL� yK/x.�Section 4 of themodel considers the government sector.We here assume,

as a slight generalization of the original Tobin approach, that moneysupplyM grows at a constant rate

�and that government expenditure is a

fixed proportion of the capital stockK (and thus also of GNPY). Taxes arethen residually determined through the government budget restraint, theGBR (2.11), since the model abstracts from bond financing of governmentexpenditures. Government savings S

�is therefore equal to the negative of

the new supply of money M� and is thus negative if this flow supply ispositive (which will generally be the case for growing economies). In thesteady state we get for the ratio t�T/K the expression g� �

�m

�, where

� Just as in the standard treatment of the depreciation �K of the capital stock K.� See appendix 2 of chapter 4 for an extension of the models of chapters 2—4, where, inparticular, wage income taxation is taken into account.

Note that later on m will denote M/(pK) due to our general choice of notation.� Note that we assume that neither firms nor workers hold money in this formulation of themodel.


m�� (M/(pK))

�which must then be constant. Taxes� (lump-sum and on

profit income solely) thus grow just as all other real magnitudes with thenatural rate of growth n in the steady state. They are (under suitable initialconditions) positive if money supply grows at a slower pace than laborsupply and negative under the reversed inequality sign, in which case theextra money flow into the economy is given as a transfer to asset holders.In section 5 of the model it is assumed that the money market (2.12) is

always cleared with respect to stock demand and supply, which by thewealth constraint (2.2) implies that the stock capital market will be inequilibrium as well. This can be thought to be achieved by instantaneousprice level changes, in the spirit of the quantity theory of money, for anygiven value of the rate of profit �. The value of the rate of profit � in turn isdetermined for any value of the price level p by the ‘‘full employment’’assumption (2.15) on the market for labor, which basically demands thefulfillment of the condition K1 � n through an appropriate choice of therate of profit � brought about by an adjustment of money wages (again forany given price level p).��It is obvious that these two conditions interact with each other and thus

jointly determine the equilibrium values of both p and � (and w via�(�)� y(1� �/x)� �).In addition to these asset market clearing conditions, equation (2.13)

states that new supply of money will always be absorbed by (asset-holding)households. First, government can of course always issue newly printedmoney by purchasing commodities with it (thereby reducing the tax pay-ments of the household sector in the presentmodel). Secondly, it is assumedas background of the equationM� �M� that households leave the decisionto reallocate their thereby increased money holdings in their portfolio tothe ‘‘next period,’’�� just accepting this additional money for the timebeing.�� The government thus can here predetermine part of the nominalflows in household savings decisions, while the remainder of it is used forreal capital accumulation and is determined in its real size by the aboveadjustments of the price level and the rate of profit. This clearly shows that

� Or, if negative, transfers.�� Note here that this full employment steady-state condition does not determine the level of

employment which, depending on initial conditions, could in principle take on any valuehere.

�� Note that there are no determining variables in the asset flow demand expressionsprovided here.

�� This simplifying assumption follows from the usual procedure of continuous-time macro-dynamics to distinguish between aWalras’ Law of Stocks and aWalras’ Law of Flows, andis here made obvious through the use of the two symbols, e.g.K� ,K� instead of onlyK� , as iscustomary in the literature (see Sargent 1987, II.7, for example). In our view, a moreintegrated treatment of asset markets is necessary — at least in a second step — to give theinvestment decision K� a more independent role. Since we, however, start from orthodoxformulations of models of monetary growth, we will not deal with such an extension in thepresent book.


the decision to invest in real capital formation is here limited in its freedomby the actions of the government.If equation (2.13) is guaranteed, it follows immediately that Say’s Law for

the market for goods must be true. But this latter consequence depends (asjust stated) on the condition that households always accept the newlysupplied money and formulate no flow demand for money that is indepen-dent of it. The consequence of this assumption is Say’s Law as formulatedin equation (2.14). That is, that the aggregated value of private and govern-ment savings must then be equal, on the one hand, to the intended rate ofchange of the capital stock K� (i.e., the investment goods demand of assetholders) and, on the other hand, to the demand gap in the market for goodsas it is described by the right hand side in (2.14) (i.e., to the supply of newcapital goods). In the present model there is thus always (at any level ofproduction) just sufficient demand to buy this production: (extra) supplyalways creates the (extra) demand for it. There are consequently no goods-market problems to be observed in this model type.The final equation of the model (2.16) describes the formation of infla-

tionary expectations � used in the definition of the rate of return differential�� as well as in the definition of disposable income Y�

�of capitalists.

Inflationary expectations can here be either purely backward looking(adaptive expectations) if �� 0 holds true or purely forward looking(towards the new steady state value

�� n of the rate of inflation) if �� 0

holds (regressive expectations); or they may be a combination of both.Groth (1988) discusses the consequences of such a mechanism of combinedadaptive forward looking expectations in the framework of IS—LM dy-namics, which will enter the stage in this book when we turn to Keynesianversions of monetary growth dynamics. Groth’s particular formulation ofthis mechanism is �� (�p� (1� �)(

�� n)��), which, however, is

easily shown to be equivalent to our formulation (2.16) by setting ��

�� and ��/��. Our approach may therefore also be interpretedas saying that inflationary expectations are revised in the light of theirdifference from a weighted average of actual short-run and actual long-run(steady-state) inflation. Of course, it may be questioned whether such anexpected rate of inflation should be used in the formation of rate of returnand disposable income expectations. Other proposals of closing the modelwith respect to expectations formation should therefore be taken intoaccount when they appear as more appropriate for the present dynamicsetup of the model. The advantage of our formulation, however, is that it isof considerable generality since it contains many cases of the literature asspecial cases (��#

� [0,�], i� 1, 2).This closes our discussion of the building blocks of our most basic model

of monetary growth. We stress again that the equations which describe thebehavior of households, firms and the government have been chosen in as


simple a form as possible in order to allow for amost transparent treatmentof the basic insights of the Tobin model of monetary growth and itsdiscussion in the literature. However, this collection of traditionally useddescriptive macrobehavioral assumptions may not yet represent the bestchoice in view of the budget restraints they are assumed to fulfill. From theviewpoint of consistency and also from the perspective of a micro-foundedpartial equilibrium analysis there may thus still exist improvements in thisstylized presentation of the assumed behavior of households in particular,which are not provided in this book. Improvements in the generality of theemployed framework will, however, be provided step by step throughoutthe book, as the discussion of the Tobin model and later of Keynes—Wicksell, Keynesian, Keynes—Metzler models, etc. proceeds. Narrow orproblematic features of the overall picture of the working of the economywill thereby be overcome to a considerable degree by the close of this book,while sectoral behavioral assumptions are kept as similar as possible inorder to allow for a proper comparison of the results obtained for thedifferent model extensions and modifications considered. Their furtherimprovement incorporating, for example, the earlier mentioned extensionto differential saving habits, more elaborate expectations mechanisms, anda more integrated treatment of asset markets will, however, here be left forfuture research.�Having constructed a monetary growth model with Tobinian as well as

Classical features we have now to proceed to the analysis of this model.This task involves the consideration of its temporary equilibrium position(the short-run analysis� as determined by full employment and money-market equilibrium), the investigation of its evolution through time (me-dium-run analysis),�� and steady-state analyses (long-run positions andtheir changes).��From a consideration of (2.2), (2.6), (2.10), (2.11), and (2.14) we find that

there are two equations in the model which determine its temporaryequilibrium positions in relative form, namely� (setting r� � r

�, i.e., equal

to its steady-state value, see below),

� See Chiarella et al. (1998) for the details of such an approach.� Of the variables �,m.�� By studying the �-dynamics in its interaction with the variable m.�� In particular, the dependence of income distribution on the growth rate

�of the money

supply.� From (2.13), (2.14)

K� �S�� S

�� s

��K�M

p� �T��

M

p� �

M�

p(by (2.6) and (2.11))

� s��K�

M

p��G�

M�

p ��M

p��

M�

p(by (2.11)).

Hence (2.17) follows by use of (2.9), (2.10) and the definition of m.


n�K1 � s�(�� g� )� (1� s

�)(

��)m,m�

M

pK, (2.17)

and, from (2.3) and (2.12),

m� h�y� h

�(r�� ), (2.18)

which can be solved for the equilibrating variables � and m, representingequilibratingmoney wage and price level adjustments via the labor and themoney markets. Note with respect to this argument that there is a one-to-one map between these latter nominal magnitudes and �,m to be derivedfrom

�� y� �� l,� �w/p,m�M/(pK),

with y, �, l� y/x and M,K being given magnitudes.Inserting (2.18) into (2.17) gives

n� s�(�� g� )� (1� s

�)(

��)[hiy� h

�(r�� )], (2.19)

from which

��s�g� � (1� s

�)(

��)(h

�y� h

�(r��) )� n

s�� (1� s

�)(

��)h

�

, (2.20)

which gives a well-defined function �(�) on the open interval(��,

�� s

�/(1� s

�)h

�) .�� This function passes through the point

(��,�

�)� (

�� n, n(1� (1� s

�)h

�y)/s

�� g� )

(due to the definition of r��

��

�), and its slope is given by

��(�)��(1� s

�)

s�

[h�y� h

�(r�� )� h

�(

��)],

as can be easily obtained from applying the implicit function theorem toequation (2.20). The slope of �(�) at �

��

�� n is therefore negative and

equal to

��(��)��

(1� s�)

s�

(h�y� h

�n).

It should be larger than �1 when empirical magnitudes are taken intoaccount. Assuming this to be true the function m defined by (2.18),

m(�)� h�y� h

�(r��(�)��) (2.21)

�� g� � n in the case s�� 1.


possesses the property

m�(��)�� h

�(��(�

�)� 1)� 0.��

The temporary equilibrium values of the rate of profit � and real balancesper unit of capital m thus both depend negatively on the unique statevariable of this dynamic model, i.e., the expected rate of inflation �. Thisconcludes the short-run analysis of the model.Turning now to the medium-run dynamics of the model, one has to start

from the inflationary expectations mechanism

�� (p��)��(�� n� �),

which gives rise to the dynamic law

�� (�� )(�� n��)� ��m,

this latter equation being due to p� �� n� m and the definition of

m�M/(pK). Note that since m is a function of � from (2.21) thenm�m�(�)/m(�)�� must hold in addition. This finally gives the autonomouslaw of motion for inflationary expectations

�� 1

1��m�(�)m(�)

(�� )(�� n� �). (2.22)

This equation has a unique steady state ��

�� n with �

��

�(��)� g� � n(1� (1� s

�)h

�y)/s

�and m

��m(�

�)� h

�y.

It is of course also possible to express the dynamic law of this basicTobin model in terms of m, via equation (2.21), which turns out to be

m�1

��(m)m��(�� )(�

� n��(m)). (2.23)

The derivative of the right hand side of the differential equation (2.22) at thesteady state is given by

��

1

1��m�(�

�)

m(��)

(�� ),

withm(��)� h

�y� 0. Since we havem�(�

�)� 0 by our above assumptions,

it thus immediately follows that there is a unique value �� of the parameter

�� m�(��)� � h

�in the special case s

�� 1.


�� below which the dynamics are stable and above which they are un-stable.�� As is well known from various formulations of Tobin’s monetarygrowth model, the steady state is thus unstable if backward looking infla-tionary expectations are sufficiently fast.Finally, concerning long-run comparisons, one has to note first that a

proper comparison demands that the reference value of the money demandfunction r

��

��

�has to be kept fixed. Otherwise, the accompanying

shift in money demand (via the switch of r�from the old to the new steady

state) would suggest superneutrality of money, since the steady-state ex-pression

��n(1� (1� s

�)h

�y)

s�

� g�

does not then depend on the rate of growth of the money supply. For a truesteady-state comparison, one has to consider the expression (2.20), how-ever, for a fixed value of r

�and ��

��

�� n. From this expression

there follows

��(��)��

(1� s�)nh

�s�� (1� s

�)nh

�

� 0,

which in the presence of smooth factor substitution would imply an in-crease in capital intensity, i.e., the Tobin effect as it is normally presented(see, for example, the discussion in chapter 1, section 1.3).�� Money istherefore not superneutral in the present model, so that both questions inthe title of this section have found a negative answer.There exist two important special cases with respect to the above dy-

namics, namely, the cases of asymptotically rational expectations ��

�(�� ) and myopic perfect foresight �� (�� ). In the firstcase the law of motion (2.22) reduces to � �

�� n, which gives stationar-

ity for the expected rate of inflation. This case shows the above-discussedTobin effect in its most basic or pure setup, i.e., as a pure comparison ofsteady-state situations without any need to consider the troublesomequestion of the stability of models of monetary growth.The second case implies via the second equation preceding (2.22)

�� p�� n� mwhich, together with the equilibrium relationshipm(�)

derived above, then gives

p� �� m(�)m�(�)

(�� n��)�

m(p)

m�(p)(

�� n� p), (2.24)

�� The system is ill defined in the borderline case. �� See also chapter 5.


due to m(�)�m�(�)�� /m(�). Given that m�(��)� 0,m(�

�)� h

�y� 0, we ob-

tain in this case of myopic perfect foresight a negative Jacobian at thesteady state, and thus always have local instability as in the case of fastadaptive expectations.Sargent and Wallace (1973) consider a partial model of equation (2.20)

and its background which exhibits the same qualitative features as thiscomplete model of monetary growth. Observing there, in a linear frame-work, global instability of their steady-state solution, they conclude thatpeople would react in such a situation by always choosing stability fromamong the possible dynamic scenarios. In the present situation, this wouldmean that an unanticipated change in the growth rate

�of money supply

��, just induces a corresponding jump in inflationary expectations

��(��

�), which would leave the system in a steady state had it been

there. The centrifugal forces of the dynamics (2.24) are thereby preventedfrom coming into operation. The motivation for this jump-variable tech-nique is that people and the public sector would not tolerate or supportpurely explosive dynamics and thus implement somehow this jump-vari-able behavior. Yet, the present model is nonlinear, so the local stabilityanalysis is not sufficient to support such a procedure. Global stabilityaspects therefore have to be taken into account before definite conclusionscan be reached. Furthermore, the price level (and the level of money wages)have been assumed to perform jumps whenever this is needed for theassumed situation of general equilibrium. This makes the temporary equi-librium part highly interdependent (even in the simple model consideredhere), and it also makes the derivation of the one law of motion that is hereallowed very cumbersome. Introducing into the framework a delayedadjustment of the price level�� (and later of money wages), via furtherdifferential equations, has the twofold advantage (as we shall see) of being(more) realistic with respect to the actual behavior of inflation rates as wellas making the temporary equilibrium part in its interaction with theassumed laws of motion much more transparent. The cost of this addedrealism and transparency is that the number of the laws of motion to beconsidered is increased thereby, necessitating the use of more sophisticatedtools of dynamic analysis or even numerical simulation.A step in this direction is provided in the section immediately following.

It will in particular show that the Sargent and Wallace (1973) jump-variable methodology is by no means compelling if global aspects of suchnonlinear monetary growth dynamics are taken into account. This con-clusion is independent of the particular price dynamics employed in thenext section which will be subject to change (in what we believe is a

�� See also Turnovsky (1995, p.76) in this matter.


direction of more empiricalmeaningfulness) from chapter to chapter in thisbook.We stress finally that the steady-state results of this section remain true

throughout this chapter.

2.2 The money-market disequilibrium extension: further stabilityanalysis

There has been a tendency to make the assumption of a sluggish adjust-ment of the price level in the face of money-market disequilibrium in thedevelopment of the literature on descriptive monetary growth models ofthe Tobin type. Such an extension of the general equilibrium version hasthe advantage of making the dynamics of the model less intertwined (byadding a second dimension to it via the assumed time lag in the adjustmentof prices to money market disequilibrium) so that the dynamic processescan be studied in a richer environment. The consequences of certainnonlinearities in the money demand function can here be studied in par-ticular by means of a new approach to the well known instability problemof such models of monetary growth.

The equations of the model are:


��w/p, u��/x,� � (Y� �K��L)/K, (2.25)

W�M/p�K. (2.26)

2 Households (workers and asset-holders):

W�M/p�K,M� h�pY� h

�pK(r� � (�� )), r� � const.

(2.27)

Y��

��K�M

p� �T, (2.28)

C� �L� (1� s�)Y��, s�

� 0, (2.29)

S��L�Y�

��C�Y� �K�T�C� s

��K�M

p� �T�

�M

p�

�M� /p�K� , (2.30)

L1 � n� const. (2.31)





M� ��M,

�� const., (2.33)

G� g�K, g� � const., (2.34)

T�G�M� /p[S��T�G��M� /p]. (2.35)

5 Equilibrium and disequilibrium conditions (asset markets):

M�M[K�K], (2.36)

M� �M� [K� �K� , see 6]. (2.37)

6 Say’s Law on the market for goods and ‘‘full employment’’:

K� � S��S

��S�Y� �K�C�G�K� , (2.38)

L�L[K1 � n]. (2.39)

7 Inflation and inflationary expectations:

p��((M�M)/(pK))� �� (1� �)(

�� n), � � [0, 1], (2.40)

�� (p��)��(�� n� �),��# � [0,�], i� 1, 2. (2.41)

This modification of the model of section 2.2 and similar variants of theTobin model have played a prominent role in the discussion of Tobin typemodels of monetary growth and their (in)stability in the literature; see inparticular the papers by Hadjimichalakis (1971a,b, 1973, 1981a,b), Had-jimichalakis and Okuguchi (1979), Hayakawa (1979, 1983, 1984), andBenhabib and Miyao (1981).It is, however, obvious that the disequilibrium approach to a theory of

the rate of inflation chosen here is at the same time very simple andproblematic. It is simple since it only adds to the model of the precedingsection the hypothesis that there will be in general a stock disequilibrium(2.36) between desired and actual money balances. This disequilibrium isthen used to determine the rate of inflation as being proportional to theexcess supply in the moneymarket� augmented by an ‘‘acceleration’’ termwhich reflects, in its first component, current inflationary expectations and,in its second component, the future steady state rate of inflation. On theone hand, this approach to inertia in the price level dynamics allows in thesteady state simultaneously for inflation and money-market equilibrium.On the other hand, it assumes (in line with certain empirical observations)

� Or equivalently, proportional to the excess demand in the market for the capital stock.


that the inertia effect of current inflationary expectations on the rate ofinflation is only partial in nature (�� 1). The advantage of this formulationof price level dynamics (2.40) in comparison to the preceding section is thatit makes the determination of the rate of inflation an explicit one, while themodel of the preceding section (which assumes with respect to (2.40) aninfinite adjustment speed �

��) must be subjected to considerable

mathematical reformulation before its determination of the rate of inflationis fully transparent. The difference here is that the price level can now onlyadjust through time while it is capable of performing jumps in the preced-ing model whenever an exogenous shock occurs in the data of the model.The simultaneous determination of the price level and the rate of profit inthe preceding model has now given way to a simple determination of therate of profit by means of the full employment condition and a delayedadjustment of the price level corresponding to the disequilibrium in themarket for money holdings.The problematic feature of this extension is that all of its assets markets

are now in (stock) disequilibrium. The theory of inflation introduced bythis extension can therefore be regarded only as a preliminary step towardsamore convincing approach to the determination of the rate of inflation (asit will be introduced in chapter 3 and more particularly in chapter 4). Notethat the new flow of moneyM� is still simply accepted by asset holders forthe time being so that Say’s Law on the market for goods remains valid inthe same way as in the preceding model.Our main interest in the following analysis of this extendedmodel is that

it allows the instability problems observed in the last section for adaptivelyformed expectations as well as for myopic perfect foresight to be treated inmore depth (and in a quite different light, by way of assuming an appropri-ate nonlinearity in the money demand function), since we have now anexplicit, though not too convincing, formulation of the dynamics of theprice level at our disposal.�In the present model, prices p are given at each moment in time and the

money wage rate adjusts instantaneously to establish the equilibriumconditionK1 � n. As in the case considered in the preceding sectionwe have(see equation (2.17))

K1 � s�(�� g� )� (1� s

�)(

�� )m, (2.42)

where m�M/(pK) and � are subject to the laws of motion determinedbelow. This last equation gives

�� g� �n� (1� s

�)(

��)m

s�

(2.43)

� To be investigated now from a global and nonlinear point of view.


as equilibrium value of the rate of profit � � y� �� (w/p)y/x, whichdetermines the unique level of money (and thus real) wages where capitaland the labor force grow at the same pace. We note that � is now anonlinear function of the two state variables m and �.To derive the laws of motion for the aforementioned two state variables,

we first express the state of the money market by means of the excessdemand function X��m� (h

�y� h

�(r�� (��) )), where we now as-

sume r�(� r� ) to be given by �

�� n and �

�by g� � n� (1� s

�)nh

�y/s

�(recall

the steady-state relationships which followed from equation (2.22)), asdefinitions of the reference rates we employ in the money demand function.Inserting the functional expression for � obtained above into this equationwe obtain

X�(m,�)� �1� h�

1� s�

s�

(�� )�m� h

�� const.,

where const. is given by �h�y� h

�(g� � n/s

�� r

�) and is �0 if �

�� 0

holds true.Due to m�

�� n� p, the dynamic law for p gives rise to our first

differential equation

m� � [�� n��

�X�� (1� �)(

�� n)]m,

� [�(�� n��)��

�X�]m, (2.44)

and for the second state variable we get from equation (2.41)

�� (��X�� (1� �)(� � (�� n)))��(�

� n��),� ((1� �)��)(� n��)� ��X�. (2.45)

The two differential equations (2.44) and (2.45) represent an autonomousnonlinear differential equation system in the variables m and �.

Proposition 2.1: Under the assumption 1� h�(1� s

�)n/s

�� 0 the

steady state of the dynamical system (2.44)—(2.45) with m�� 0 is unique

and given by

m�� h

�y, �

��

�� n.

The steady state values of r and � are therefore exactly the ones we haveused as reference in the money demand function m.

Proof: Setting equations (2.44) and (2.45) equal to zero gives thefollowing linear equation system for

�� n��,X�.

��m ��

�m

(1� �)�� n� �

X� ��0

0� .


The determinant of the matrix on the left hand side is positive (m�� 0!),

implying only a zero solution of this equation system. Thus ��

�� n

and m� h�y� h

�(r��

�), where r

�is given as defined above — based

on the expression

�� g� �

n� (1� s�)nh

�y

s�

for the rate of profit. On the other hand proceeding directly from (2.43) wehave

�� g� �n� (1� s

�)nm

s�

.

Due to our choice of the reference rate r� in the money demand function wetherefore know that m

�� h

�y can fulfill these equations simultaneously,

and thus provides a steady-state solution for the dynamics. Furthermore,the above functional expression forX�(m,�) can be uniquely solved for m

�,

(due to ��

�� n), implying that there can be no further steady state

solution of the dynamics (apart from the one that corresponds to m��

0).�

Proposition 2.2: Assume s�� 1/2, h

�y� 1 and �

�� (1� �)/

X��(� 0). There then exists exactly one value �� of the parameter ��wherethe system (2.44)—(2.45) undergoes a Hopf-bifurcation by switching fromlocal asymptotic stability to local instability as �� passes through �� frombelow. Generally, this loss of stability is accompanied by either the death ofan unstable limit cycle or the birth of a stable limit cycle (periodic motion)as the bifurcation parameter �� is crossed from the left.

Proof: The Jacobian of the dynamical system (2.44)—(2.45) at thesteady state reads

J��

�X��m

��

�X��m�

� �m�

��X��

��X�� (1� �)�� .This gives for the determinant of J the expression

det J��(�� )X�

�m

�,

which is always positive, since the steady-state value of X��

�

1� h�(1� s

�)/s�n is positive for all n� 0.

On the other hand for the trace of J we obtain

trace J� � (1� �)�� X��m

�� X��,


with

X�� h��1�m

�

1� s�

s�� 0,

if the (empirically plausible) restrictions m�� 1, s

�� 1/2 are assumed.��

Furthermore ��X�� 1� � by the assumption on the parameter �

�. For

any given parameter values ��,�� (with �

�X�� 1� �) we therefore get a

positive number

��X��m

��

��X�� (1� �)

,

at which trace J becomes zero (it is negative for �� and positive for��). In view of this simple situation with respect to trace J and det J,the proof of the Hopf bifurcation is now a routine exercise; see, e.g.,Benhabib and Miyao (1981) for the details in a related, but three-dimen-sional situation (the third dimension arising due to their use of smoothfactor substitution).�

In respect of this bifurcation result we note that for � � 1, �� 0, theabove bifurcation value reduces to �� X�

�m

�/X��which is formally close-

ly related to the so-called Cagan (1967) stability condition and the Tobin(1975) stability condition on ��(��), see Groth (1992, 1993) for details.Similar conditions for local asymptotic stability will reappear in laterchapters of this book. Furthermore, lower parameter values of � and highervalues for �� increase the stability domain, while the influence of �

�is

ambiguous. Whether the Hopf bifurcation that has been shown to occur issupercritical, subcritical or degenerate�� has to be determined by numeri-cal simulations of the model, due to the analytical difficulties involved inthe calculation of the Liapunov coefficient (see Kuznetsov 1995) whichdistinguishes between these three cases. Though the loss of stability at �� isnecessarily accompanied by a cyclical reaction pattern, such cycles willdisappear (giving rise locally to monotonic explosiveness) when the par-ameter �� is further increased. This is due to the fact that it enters bothtrace J and det J in a linear fashion, so that �� (trace J)�/4� det J mustbecome positive for sufficiently large ��, implying real eigenvalues for theJacobian of the dynamics at the steady state.The genesis of instability is the autofeedback chain

��X��p��

�� As the expression for the trace shows, the partial derivativeX��(X��) plays a similar role astheKeynes effect (Mundell effect) in IS—LMmodels, but now in a Tobin (1965) typemodel.

�� Giving rise to stable or unstable limit cycles or center type dynamics, respectively.


and is based on the positive reaction of the excess supply of real balanceson expected inflation. As shown, this reaction pattern becomes the domi-nant one, if �

�is larger than (1� �)/X�� and �� is chosen sufficiently large.

Let us next calculate the isoclines of the above two-dimensional system.These turn out to be

�(�� n� �)��

�X�, (from m� � 0),

[(1� �)�� ](�� n� �)��X�, (from �� 0).

Inserting the expression for X� we have derived above then gives respect-ively�

m�(�(

�� n� �))/�

�� h

�� const.

1� h�(1� s

�)(

��)/s

�

, (from m� � 0),

and

m�[(1� �)�� ](�� n�

�)/(��)� h

�� const.

1� h�(1� s

�)(

��)/s

�

,

(from �� 0).

These two curves intersect at the steady state (m�, �

�). The first isocline is

below the second to the right of this steady state (��) and above it to its

left (�� ).

For the remainder of this section we assume for simplicity s�� 1. This

gives for m� � 0

m� �(�� n��)/�

�� h

�� const.,

indicating that m is a decreasing function of �. For �� 0 we get

m� [(1� �)/��/(��)](�� n�

�)� h

�� const.,

which is again a linear function of � and decreasing (increasing) accordingto

(1� �)/��/(��)� h

�� 0(� 0).

Assuming values of ��such that (1� �)/�

�� h

�� 0 holds again gives that

sufficiently large �� will imply a decreasing �� 0 isocline and a locallyunstable steady state of the dynamics (2.44), (2.45).Let us assume that this situation prevails at and around the steady state.

Due to the wealth constraint (2.3), it is obvious thatW�M/p� 0 shouldbe fulfilled by the demand function for real balances. The postulated linearshape forM/p can therefore be true only for a certain neighborhood of thesteady state. Far away from the steady state, nonlinearities in this demand

� const.� h�(g� � n/s

�� r

�)� h

�y� 0 [if �

�� 0].


Figure 2.1 Simple nonlinear money demand function

function must necessarily arise. A simple and still preliminary formulationin the case s

�� 1 (��

�) is given by the assumption of the dependence of

the money demand function parameter h�on � displayed in figure 2.1.

The shape of the h�( · ) schedule implies that h

�must approach zero for

large positive as well as negative values of inflationary expectations. Thesensitivity with respect to the rate of return differential is therefore decreas-ing the further away this differential is from its steady state value 0. Westress that this nonlinearity is chosen for expositional reasons solely. Assetmarkets are not the central theme of the present book (due to its method ofmodel progression), but in future research theywill have to be reformulatedin a way that we sketch in the last chapter of this book.��The phase diagram in figure 2.2 indicates how the dynamics change

when monetary disequilibrium growth takes account of the money de-mand nonlinearity in figure 2.1.�� The arrows in this figure indicate thedirection of motion off the isoclines. In view of these adjustment directionswe have drawn a rectangle (beginning with point A) that is obviously aninvariant set of these dynamics, since the horizontal line cannot be crossedby any trajectory. This figure therefore suggests as a routine application ofthe Poincare—Bendixson theorem the following proposition.

�� Then, in particular the pK component of money demand must be replaced by pW, leadingto a greater interdependence between the real and the financial sectors of the economy.

�� See Chiarella (1990, ch. 7) for another approach of this kind in the case�� 0, �

��

�� n� 0.


Figure 2.2 Bounded fluctuations for disequilibriummonetary growth

Proposition 2.3: For the dynamical system (2.44)—(2.45) incorpor-ating the money demand nonlinearity of figure 2.1 there exists at least one(attracting) periodic motion in the domain ABCD.

We note that this periodic motion is implied by global aspects and not bylocal ones as was the case in proposition 2.2.

Proposition 2.4:Assume �� 0. Then, the isoclines of the dynami-cal system of proposition 2.3 are independent of the size of �� and thesystem tends to relaxation oscillations as the parameter �� goes to infinity.

If �� is increased towards infinity the dynamics therefore approach thesituation of a relaxation cycle as depicted in figure 2.3. Therefore, despitethe fact that we get a monotonically explosive situation around the steadystate as �� (as shown above), we will always have a unique andattracting monetary growth cycle in this model when viewed from a globalperspective. This situation will also hold true if �� 0 is assumed, giventhat the �� 0 isocline converges to the isocline shown in figure 2.3.We thus arrive at the conclusion that Tobin type (local) monetary

growth instability will in fact generally give rise to viable limit cycledynamics if global aspects are taken into account. Note, however, that thismay not be the case if the forward-looking elements in the price dynamics,(1� �)(

�� n), and the expectations mechanism, ��(�

� n��), are sup-pressed (as is often done) by assuming �� 1 and at the same time �� 0.


Figure 2.3 The case of relaxation oscillations or limit limit cycles (for �� 1,��

In this case, the �� 0 isocline is given by m�� h�(�)�� const. and does

not give rise automatically to the upward sloping parts we have used aboveto show the existence of a stable limit cycle and the viability of theconsidered dynamics.Apart from this latter case, however, we have that the unstable steady

state is surrounded by persistent fluctuations in real balances per unit ofcapital m and inflationary expectations �, where m (and p) is moving in acontinuous fashion, while inflationary expectations jump at certain points intime in order to establish a new perfect foresight equilibrium p� �, witheither a high rate of inflation which will start declining after this jumpoccurred or a low, but then rising, rate of inflation. Both of these continu-ous movements are terminated after a certain amount of time when an-other jump to the opposite situation again occurs. The limit case��(�� p) of perfect foresight thus here is of quite a different nature incomparison to the jump-variable technique (for p) commonly used inmacroeconomic theory since Sargent and Wallace’s (1973) basic formula-tion of it. This is due to the fact that the global point of view adopted hereautomatically limits the explosive situation to regions around the steadystate. It was the assumption that this explosive behavior was global thatwas used by Sargent andWallace to justify their argument that unforeseenshocks in the money supply could only lead to jumps in the price level p, sothat the endogenous variables (m,�) just remain in the steady state at such acase.


Sargent and Wallace not only assume ��(p��), but also��(m� h

�y� h

�(r�� ) with � fixed at a given value). In this

case, the above two isoclines become identical (equal to m�

� h�� const.). The dynamics then again become of the type considered

in the preceding section if one considers this limit case without payingattention to its neighboring case of a very high, but finite adjustment speedof prices. If compared with these neighboring cases, however, we see thatthe limit cycle considered in figure 2.3 simply grows in amplitude (inthe �-direction) towards infinity as �

��. This means, on the one hand,

that the model is not yet well defined (that is to say viable) for ��

and, on the other hand, that the resulting limit limit limit cycle(�� , etc.) again suggests a difference to theSargent and Wallace jump-variable technique. We conclude that suchmodels of monetary growth (as well as the ones of subsequent chapters)should be further extended and be analyzed taking account of their intrin-sic nonlinearities as well as further extrinsic ones. Such analysis will furthersupport the conclusion that their dynamical behavior is more likely toexhibit persistent fluctuations around the steady state rather than to be ofthe stable saddlepath type converging toward it that is generally employedin the literature.This latter stable saddlepath dynamics is of a purely forward looking

behavior where agents do not take into account any experience obtainedfrom the past up to the present date. Our approach, by contrast, combinesbackward and forward looking elements (�� 1, �� 0) in a still simpleway in order to obtain quite different conclusions from those of themonetary growth dynamics of the neoclassical variety. Now, myopic per-fect foresight is made compatible with persistent fluctuations if the adjust-ment speed of the price level is finite, so that only inflationary expectations� can (and quite sensibly will) perform jumps at certain points in time.�

2.3 Labor-market disequilibrium and cyclical monetary growth

We now extend the model to include labor-market disequilibrium and a(somewhat) sluggish adjustment of nominal wages in the light of suchdisequilibrium. This immediately adds two new laws of motion to thedynamics we have discussed so far, namely, the resulting real wage dynam-ics and the now generally fluctuating growth rate of the capital stock. Theinteraction of these two dynamic variables has been extensively studied inmodels of cyclical growth of the Goodwin and Rose type. The questionwhich then arises is in what way will the overshooting cyclical features of

� See Flaschel and Sethi (1999) for a more elaborate treatment of this topic.


these real growth models reappear in such a higher dimensional model ofmonetary growth?The equations of the model are:


��w/p, u��/x,� � (Y� �K� �L)/K, (2.46)

W�M/p�K. (2.47)

2 Households (workers and asset holders):

W�M/p�K, M� h�pY� h

�pK(r� � (��) ),

r� � const., (2.48)

Y��

��K�M

p� �T, (2.49)

C��L� (1� s�)Y��, s�

� 0, (2.50)

S��L�Y�

��C�Y� �K�T�C� s

��K�M

p� �T�

�M

p�,

�M� /p�K� , (2.51)

L1 � n� const. (2.52)




M1 ��, (2.54)

G� g�K, g� � const. (2.55)

T�G�M� /p[S��T�G��M� /p]. (2.56)

5 Equilibrium and disequilibrium conditions (asset markets):

M�M[K�K], (2.57)

M� �M� [K� �K� , see 6]. (2.58)

6 Say’s Law on the market for goods and labor-market disequilibrium:

K� � S��S

��S�Y� �K�C�G�K� , (2.59)


L�L[V�L/L� const.]. (2.60)

7 Wage–price inflation and inflationary expectations:

w� ��(V�V� )�

�p� (1�

�)�, (2.61)

p��((M�M)/(pK))�

�w� (1�

�)�, (2.62)

�� (p��)��(�� n� �),��# � [0,�], i� 1, 2. (2.63)

The model of the preceding section 2.2 is here extended to allow for thetreatment of labor-market disequilibrium (2.60), where it is assumed thatthe rate of employment V is now varying around its so-called natural levelV� (� 1), which, as in most macroeconomic treatments of the so-calledNAIRU rate of employment, is given exogenously (see however chapter 7for an exception).We assumehere, as well as in all othermodels that follow,that the maximum value of the rate of employment V (which includesovertime work�) is considerably larger than 1 and that this absoluteceiling to the employment of the labor forces is never reached by themedium-run dynamics investigated in this book. This is an assumption orrestriction which helps to avoid regimes of so-called repressed inflation, anassumption which may be justified by a very high degree of wage flexibilitynear absolute full employment such that this flexibility induces the dynami-cs to stay away from absolute full employment. Note that our formulationof Say’s Law (2.59) does not depend on the fact that there is now labor-market disequilibrium present in our Tobin type model.�Here, however, we shall remain with linear economic mechanisms and

thus defer the consideration of the aforementioned nonlinearity close toabsolute full employment to later investigations. Our wage adjustmentequation (2.61) therefore does not yet allow for the treatment of suchconjectured stabilizing mechanisms. Wage inflation w is here strictly pro-portional to deviations of the rate of employment V from its NAIRU levelV� (which need not be determined by the so-called NAR, the natural rate ofunemployment). Wage inflation is furthermore assumed to be influencedby the observed actual rate of price inflation p as well as the expectedmedium-run rate of inflation �, which is assumed to follow some average ofthe short and the steady rate of inflation, p and

�� n. A similar assump-

tion is made for the determination of price inflation (2.62), which modifiesthe formula used in the preceding section. Note that the use of (2.62) (and of(1�

�)�) in the place of (2.40) (and there of ��) explains from another

perspective why empirical observations on the value of � generally report a

� See the model of chapter 7 for a more detailed treatment of this aspect.� It is easy to establish that Y� �K�C�G�K� must hold true in this model type at all

moments in time independently of the state of the labor market.


value of less than 1. Note, furthermore, that the term �w incorporates the

usual procedure of static markup pricing (a cost-push term) into thedeterminationof price inflation. Note, finally, that we have already stressedthat the term (M�M)/(pK) may not represent a convincing expressionfor all remaining influences on the rate of inflation (the demand pressurefactors).The formal structure of our wage and price adjustment equations is the

same as that used in Rose (1990), to whom we owe this idea of representingthe wage—pricemodule of the model, though its interpretation and applica-tion differ considerably from that of Rose.Note that these wage and priceadjustment equations can also be represented in the following form:

w� ��(V�V� )�

�(p� �),

p�� ((M�M)/(pK))�

�(w��).

Represented in this format they can be interpreted as follows. Deviations ofprice and wage inflation from the expected medium-run evolution ofinflation � as such are caused on the one hand by disequilibria in themoney market or in the labor market (representing the demand-pressurefactors) and deviations of actual wage or price inflation from the expectedrate � (representing cost-push factors). These equations can therefore beconceived as a considerable generalization of many other formulations ofwage and price inflation (though of course the formulation of one of thedemand pressure terms needs further improvement as discussed earlier).The above two equations are linear equations in the unknowns

w� �, p��, which are easily solved (on the basis of the assumption1�

� �� 0 which is made throughout this book). They give rise to the

following expressions for these two unknowns ( � (1� � �)��):

w� �� [��(V�V� )�

��((M�M)/(pK))], (2.64)

p�� [ ��(V�V� )��

�((M�M)/(pK))], (2.65)

which in turn imply for the dynamics of the real wage � �w/p,

�� w� p� [(1�

�)��(V�V� )� (1�

�)��((M�M)/(pK))]. (2.66)

These representations of the dynamic laws that govern the nominal and thereal magnitudes of the wage—price module will be often used throughoutthis book and its various model types. This completes the description of

See also Solow and Stiglitz (1968) for a similar approach to the wage—price level interac-tion.

Since the magnitudes ofM� /p and of w, � are predetermined at each moment of time, it ishere demanded that S

�always fulfills the side condition S

��M� /p�� K.


the innovations in the present variant of the Tobinmodel type (see sections2.1 and 2.2 for the description of its other, unchanged components).There are now four state variables of the model and four corresponding

laws of motion which form an autonomous system of differential equationsof dimension four. These laws read

�� [(1� �)��X�� (1�

�)��X�], (2.67)

l1 � n� s�(�� g� )� (1� s

�)(

��)m, (2.68)

m� �� n�� [

��X��

�X�]� l1 , (2.69)

�� [ ��X��X�]��(�

� n��), (2.70)

where we have made use of the abbreviations,

X��V�V� ,V� l/l, l� y/x,X�� (M�M)/(pK)�m�m,m� h

�y� h

�(r�� ),

�� y(1� �/x)� �.

Note for future reference that

X��y

lx�V� ,

X��m�h�y

x� � h

�� h

�y� h

�(r�� y� �).

Equation (2.67) follows directly from (2.66) and the definitions of X� andX� above. Equation (2.68) follows from l1 � n�K1 and the expression forK1which is obtained from manipulations similar to those that lead to equa-tion (2.17). Equation (2.69) is obtained from m�

�� p�K1 �

�� n�� (p��)� (n�K1 ) by making use of (2.65) and (2.68). Equa-

tion (2.70) follows from (2.63) and (2.65).The unique interior steady state of the dynamical system (2.67—2.70) is

the same as the one obtained in section 2.2, but in addition now includesthe equilibrium conditionsV�V� ,m�m.�Thus we can write the steady-state values as

�� (y� ��

�)/l, l� y/x (2.71)

��n(1� (1� s

�)m

�)

s�

� g� , (2.72)

� In the proof of this assertion, one has to make use of the two equations p� � �0 and�� 0 as simultaneous linear equations for X� and X�, which imply X�� 0,X�� 0 astheir only solution.


m�� h

�y, (2.73)

��

�� n, (2.74)

r��

��

�, (2.75)

l�� y/(xV� ). (2.76)

Note that we have again made use of r�in the place of r� in the money

demand function m, which simplifies the calculation of the steady state.Taking this into account the non-superneutrality of money follows again,as in section 2.2.Let us consider two polar subcases of the full dynamics first. One subcase

(the real cycle) is determined by �

� 1 and s�� 1, which gives the auton-

omous subdynamics for � and l,

�� X�, (2.77)

l1 � n� (�� g� ), (2.78)

where X�� y/(xl)�V� ,�� y(1��/x)� �. The real cycle implied by(2.77)—(2.78) is a standard formulation of the Goodwin (1967) growth cyclemodel. It therefore gives rise to closed trajectories solely.The other subcase (the monetary cycle) is given by

�� 1, �

�� 0,

��, and l� l

�(l1 � 0), and results in the two autonomous dynamic laws

for m and �,

m� �� n��

�X�, (2.79)

�� X�� (�� n��), (2.80)

where X� is given by m� h�y� h

�(r��

��). This model is but a

special case of the two-dimensional model of section 2.2 (with �� and

�� 1 basically). The propositions of section 2.2 also apply here (thoughwith less stringent assumptions), which justifies the designation ‘‘monetarycycle’’ of this submodel. Note here that local instability is again due toX�� 0 (i.e., a positive feedback of expected on actual inflation), whileX�

�is

a stabilizing influence.� These two arguments both apply to the trace ofthe Jacobian J of the system (2.79), (2.80), since det J� 0 is always true.Having obtained two independent cycle generating mechanisms as sub-

systems of the full four-dimensional dynamics (one new, one known fromsection 2.2), we now proceed to a preliminary investigation of their interac-tion. It is known from the literature, (cf chapters 1 and 3 of this book), thatthe center-type stability of the Goodwinmodel can be made explosive if anappropriate price-dynamics mechanism is added to it, as, for example, in

� A negative feedback of the price level on the rate of inflation.


Rose (1967). Global stability is then obtained via smooth factor substitu-tion and ever increasing wage flexibility when the system departs furtherand further from its steady state. This Rose mechanism is also present inthe current framework, but due to the more general structure it is not nowpossible to disentangle the real and monetary mechanisms as in Rose(1967) and related approaches. The issue of analyzing separately the realand monetary sectors in such a framework is taken up in chapter 3.To see the above point in a basic format, assume again s

�� 1, but now

set �� 1 so as to switch on the interaction of the price dynamics with the

real cycle. Furthermore, �� 0(��) is assumed so as to allow us toignore the � dynamics for the time being. We then get the following fullyinterdependent three-dimensional dynamical system

�� [(1� �)��X�� (1�

�)��X�], (2.81)

l1 � n� (�(�)� g� ), (2.82)

m� � [ ��X��

�X�]� l1 , (2.83)

with X�� yl/x�V� ,X��m� (h�y� h

�(r�� (�)�

�� n)),�(�)�

y(1��/x)� �.The Jacobian of the dynamical system (2.81)—(2.83) at its steady state is

given by

J�� (1�

�)��

�h�y/x � (1�

�)��

�y/(xl�

�) � (1�

�)��

�l�y/x 0 0

( ��h�� 1)m

�y/x

��m

�y/(xl�

�) � �

�m

�� ,

(2.84)

which has the sign structure

J��

� 0 0

� � �� , (2.85)

sinceX��(� � h�y/x)� 0,X�

�(� 1)� 0 hold. Therefore det J� 0 and the

principal minors J,J

�satisfy J

� 0, J

�� 0. Furthermore J

�(� (1�

�)��

�m

�y/x)� 0 by a simple explicit calculation of this principal minor

of J. For trace J we have

trace J� � (1� �)��X��

�X��m.

The first term in this trace represents the destabilizing Rose effect, statingthat a declining real wage (due to a rising price level) will induce a further


rise in the price level and thus further declining real wages and so on. Thesecond term in the trace is the stabilizing effect that price level increasesexercise on the rate of price inflation via a reduction of the real moneysupply. The above partial derivatives read explicitly

X�� h�y/x,X�

�� 1.

The Rose effect will therefore dominate the trace of J if h�is large and thus

interest rate flexibility low. This indeed resembles the assumptionsmade byRose (1967) in the Keynes—Wicksell framework in order to obtain localinstability for the steady state. The Rose mechanism therefore operates insimilar fashion in the framework of neoclassical monetary growth.

Proposition 2.5: The monetary growth model (2.81)—(2.83) is lo-cally asymptotically stable if �

�and h

�are chosen sufficiently small.

Proof:The role of h�in trace J has already been considered above.

It remains to be shown that the coefficients a�, a

�, a

of the characteristic

polynomial of J (which are already all positive) fulfill the Routh—Hurwitzcondition b� a

�a�� a

� 0, where a

�� trace J, a

��J

�� J

, a

�

�det J.To show this last condition for local asymptotic stability, we note that

�� 0 implies a

� 0 (since � J � � 0), while a

�(for h

�sufficiently small) and

a�stay positive. Therefore, b� 0 for �

�sufficiently small.�

Proposition 2.6: The monetary growth model (2.81)—(2.83) under-goes a Hopf bifurcation as the parameter h

�is increased, starting from a

position of local asymptotic stability as described in the preceding proposi-tion.

Proof: The result is easily established by considering the effect ofan increasing h

�on trace J, see Benhabib and Miyao (1981) for the details

of the proof strategy.�

Loss of stability therefore comes about in a cyclical fashion, accompaniedby birth or death of certain periodic motions. Note that the role of �

�and

��with respect to such a loss of instability is not as obvious, as these

parameters have no influence on the sign of trace J, but have to beconsidered via their influence on the b term of the Routh—Hurwitz condi-tion in proposition 2.5. We here can only state that large values of �

�will

See pp.67—68 for a summary of details of the Routh—Hurwitz condition for three-dimen-sional dynamical systems.


imply stability, when trace J� 0 is fulfilled, since a�a�$��

�and a

$ �

�hold. Note finally that the model is always locally asymptotically stable ifk�� 1 is again assumed (the Goodwin subcase).We have so far assumed s

�� 1. The case s

�� 1 implies extending the

dynamics with the differential equations for l,

l1 � n� s�(�(�)� g� )� (1� s

�)nm, (2.86)

generated via the concept of disposable income of households that isemployed in the present type of monetary growth model. The Jacobian ofthe resulting dynamical system is obtained by adding to the Jacobian (2.84)the matrix

�0 0 0

0 0 (1� s�)nl

�0 0 (1� s

�)nm

�� .

These are, however, minor additions to the structure so far analyzed thatwill not alter the qualitative dynamic behavior already obtained. Wetherefore conclude that the concept of disposable income of these Tobintype models does not influence its stability properties in a significant way.The same is, of course, true with respect to the model of section 2.2.Let us now consider the general four-dimensional case (2.67)—(2.70).

With respect to the Jacobian J of this system at the steady state� we canstate:

Proposition 2.7: The Jacobian J (at the steady state) of the fullfour-dimensional dynamical system (2.67)—(2.70) satisfies � J � � 0.

Proof: Note first that the Jacobian in question is given by

J�� (1�

�)��

h�y

x�

�� (1�

�)��

V�

l�

��

� (1� �)��

�� (1�

�)��h��

�

s�V� 0 l

�(1� s

�)(1��

�)n �l

�(1� s

�)(1��

�)m

�

m��

�

h�y

x�s�V�

l� m

�

��

V�

l�

m��

�� (1� s

�)(1��

�)n� �m

��(1��

�)(1� (1� s

�)m

�)� �

�h��

�� h�y

x��

�

V�

l�

�� h�� .

To prove the proposition one has to note first that there are many lineardependencies present in this Jacobian which, when removed by elementary

� We note the following derivatives at the steady state:

X��X��

�X�� 0,X��

�� y/(xl�)�,

X�� h�y/x,X�

�� 1,X�� h

�,X�

�� 0.


row and column operations, make this proposition obvious. Thus det J canbe reduced to�

�0 � (1�

�)��

V�

l�

��

��

� �

0

s�V� � l

�(1� s

�)(1��

�)nh

�y

x0 l

�(1� s

�)(1��

�)n �l

�(1� s

�)(1��

�)(m

�� h

�n)

0 0 0 �m�(1��

�)�

m��

��0 ��

�

V�

l�

0 ��

from which the assertion

det J�m��

� ��1� �

��

��

V�l�

�s�V� � l�(1� s

�)(1��

�)nh

�y

x �� 0

follows.�

Proposition 2.8: The steady state of the four-dimensional system(2.67)—(2.70) is locally asymptotically stable if �

�, h

�,�� are chosen suffi-

ciently small.

Proof: The case with �� 0 [�� arbitrary] has already beenproven above to be locally asymptotically stable when h

�and �

�are

chosen sufficiently small, since the system then separates into independentlocally stable three-dimensional dynamics (with three eigenvalues havingnegative real parts), and an appended one-dimensional differential equa-tion with a negative (zero) eigenvalue if �� (� )0. Since the eigenvaluesdepend continuously on the parameters of themodel, the eigenvalues of therelated four-dimensional case with �� 0 and sufficiently small must alsohave negative real parts throughout.�

The trace of the Jacobian of the four-dimensional case at the steady state isnow given by

trace J�� (1� �)��X��

�X��m��

�X�� ,

� ��h�[(1�

�)�y/x��]�

� This simplification has been achieved by the following set of elementary row and columnoperations

col. 1�h�y

xcol. 3, col. 4� h

�col. 3, row 3�

m�

��row 4,

col. 3��l�

�� V�col. 2, row 3�

m�l�

row 2.


m(1� s�) (1��

�)n� ( �

�m��).

Continuing the arguments on the trace in the three-dimensional subcaseconsidered above we can therefore state (X�� 0) trace J� 0 iff h

�and �

�(or ��) are sufficiently large for given values of the components of the trace.Price flexibility (for sufficiently large h

�) and fast adaptive expectations

thus work against local asymptotic stability, due again to the Rose effecttogetherwith the Cagan effect, on the one hand, and the Cagan effect alone,on the other hand (in the real and the monetary part of the model,respectively).Note here that �

�does not play a role in the size of the trace. This will

not be so in subsequent chapters. Note again, that the Tobin effect entersthe Jacobian in such a way as to avoid its amplification by appropriateparameter choices. Instability caused by inflationary expectations is there-fore primarily due to the Cagan effect (based on the asset demand substitu-tion process) rather than to the Tobin (disposable income) effect (in the cases�� 1), since X�� h

�� 0 and X�

�� 1� 0.

Summarizing, the four-dimensional case represents a synthesis of Good-win’s growth cycle model with a two-dimensional cycle of Cagan type inthe monetary part of the model, augmented by a Rose effect in the firstcomponent of the trace (X�� 0). These are cycle generating mechanismswhich are coupled through the dependence of X� on m and � (the Caganand the Tobin effect), on the one hand, and the influence of X� and X� onthe rate of inflation (and on l1 on m), on the other hand. Due to the abovepropositions, cycles in the four-dimensional case will arise via Hopf bifur-cations if �

�, h

�, and �� are chosen appropriately, i.e., in particular the

system can lose stability only in a cyclical fashion. Note here that economicfour-dimensional examples of Hopf bifurcations are rare in the literatureand that alternative methods, in particular global ones, of proving cyclicalphenomena in this dimension will in general demand very special situ-ations.Let us, finally, compare briefly the above Goodwin—Rose extension with

the basic format of the Tobin model of the literature. To obtain this lattermodel type, one has to assume s

�� s

�� s[G� 0, � � 0], and �

��

��

�� , and add smooth factor substitution. There is thus already aconsiderable difference to be noted between these two versions of neoclas-sical monetary growth theory.

2.4 General equilibrium with a bond market: concepts of disposableincome and Ricardian equivalence

In this section we extend the general equilibrium version of the Tobinmodel by including government debt and bonds. This has the advantage


that monetary policy can now be separated in a clearer way from fiscalpolicy and it puts the question of the (non)superneutrality of money into amore adequate environment, since additional money supply (through ahigher rate of growth) can, in principle, now be exercised through openmarket policies and thus no longer implies changes in either taxes orgovernment expenditures (i.e., changes in fiscal policy). Furthermore, thepresence of bonds implies that there now exist competing formulations fordisposable income, depending in particular on the answer that is given tothe question: Are government bonds net wealth?The equations of the model are:�


��w/p, u��/x,� � (Y� �K� �L)/K, (2.87)

W� (M�B)/p�K, p%� 1. (2.88)


W� (M�B)/p�K,M� h�pY� h

�pK(1� �)(r� � r), (2.89)

Y��

��K� rB/p�M�B

p��T, (2.90)

C��L� (1� s�)Y��, s�

� 0, (2.91)

S��Y��C�Y� �K� rB/p�T�C� s

�Y��

�M�B

p�,

� (M� �B� )/p�K� , (2.92)

L1 � n� const. (2.93)




G�T� rB/p��M/p, or

G�T� (r��)B/p��M/p [see (2.96)], (2.95)

� This model as it is formulated below, at first makes use of the conventional Hicksiandefinition of perceived disposable income (here of asset holders solely) Y�

��K� rB/

p�M�B

p� �T, which is to be distinguished from actual disposable income

Y�� K� rB/p�T (see Sargent 1987, p.18). A prominent alternative definition for Y��is the Barrovian one, which in the present context is given by Y�

��K� rB/

p�M

p��

B�

p�T. This definition will give rise to Ricardian equivalence also in the

present model as we shall see below. See also Sargent (1987, ch. I.10).


T� �(�K� rB/p), (2.96)

S��T� rB/p�G[� � (M� �B� )/p, see below], (2.97)

M1 ��, (2.98)

B� � pG� rB� pT�M� [� (��

�)M]. (2.99)


M�M� h�pY� h

�pK(1� �)(r� � r), (2.100)

B�B[K�K]: (1� �)r� �� (1� �)�, (2.101)

M� �M� ,B� �B� [K� �K� , see 6]. (2.102)

6 Say’s Law on the market for goods and ‘‘full employment’’:

K� � S��S

��S�Y� �K�C�G�K� , (2.103)

L� const. ·L[K1 � n]. (2.104)

7 Adaptive, regressive, or perfect expectations:

�� (p��)��(�� n� �),��#

� [0,�], i� 1, 2. (2.105)

The model presented above returns to the general equilibrium formulationof our most basic Tobin model in section 2.1, but extends this model byintroducing bonds and interest rate phenomena.In section 1 of this formulation we have therefore added, in comparison

to the gross real rate of return on capital �, the gross real rate of interestr�� and a definition of real wealthWwhich now includes bonds issued bythe government. These bonds are of the fixed-price variety (with pricep%� 1) and varying nominal interest payments r per bond (r� the nominal

rate of interest). In assuming this type of bond we follow Sargent (1987,p.12), who states that this asset can be viewed essentially as a savingsdeposit. This is done solely for reasons of comparison and analyticalsimplicity, and will surely demand change once the stage is reached whereproblems of the financial sector are analyzed more thoroughly. Yet, in thisbook, we will make use of this type of government asset solely and will addequities (as perfect substitutes for such bonds when the stage of an indepen-dent investment behavior and the financing of it is reached; see chapter 3).Throughout the book, the asset structure will therefore remain a verysimple one, in fact one that should not give rise to severe economicproblems of the type discussed under the heading ‘‘financial fragility,’’ andthe like. The financial side of all of our models thus only prepares theground for future research on more problematic asset structures, imperfectsubstitutability, etc., whose results will allow a useful comparison with


those obtained in this book for this simplest setting of the financing ofgovernment expenditures (and later on also of firms’ investment expendi-ture).Section 2 of the model integrates bond holdings and bond demand into

household behavior. Looking at the wealth constraint we have now threecomponents of asset demandwhere the demand formoney is essentially thesame as in our former models, but now based on the nominal rate ofinterest in place of the former rate of return differential betweenmoney andreal capital. Note also that we now allow for a rule of tax collection bymeans of a given tax rate �, which means that we have to employ the netrate of interest (after taxes) in the money demand function.In accordance with our definition of real wealth we have to revise the

concept of disposable income as perceived by asset holders employed thusfar. It must now also include a term (B/p)� that refers to the purchasingpower losses of real bond holdings. Furthermore, asset owners now receiveinterest payments besides profits, which have to be added to their (per-ceived) disposable income. These are the necessary changes in equations(2.91) and (2.92) with respect to income accounts. The intended allocationof savings in (2.92) must, of course, now also include bonds as the alterna-tive for asset holders.There is at present no change necessary in the description of firms. The

government sector, however, is to be reformulated in a significant way toinclude a consistent description of its debt financing. Taxes (2.96) are nowbased on profits and interest payments and assumed to be a constantfraction of them.�Note that the government expenditure equation (2.95) isdifferent from the rule we have employed so far. It is based on taxes,augmented by a term

�M/p which takes account of the fact that govern-

ment should continuously supply extra money in a growing economy andspend it on extra government purchases of goods, here to the extentdescribed by the parameter

�. Note also that the most appropriate par-

ameter value for �is given by n, the natural rate of growth, implying that

government debt will then be zero in the steady state, as we shall see lateron.�Government savings (2.97) are in the present model additionally reduced

by the interest payments on outstanding debt (if B is positive), and they cannow be financed (if they are negative) by additional money as well as bondsupplyM� ,B� . Since the magnitudes of G, T, and M� are all determined bypolicy rules, the amount of new debt financing is determined passively in

� Taxes on wage income are here excluded solely for reasons of simplicity, not because this ismore appropriate; see appendix 2 of chapter 4 for the general case.

� It will be positive and growing in the case �� n and negative (describing a creditor

position) in the opposite case.


this model and has to be calculated from the government budget restraint(GBR), as given in equation (2.99). There is one new equilibrium conditionnow present, i.e., equation (2.101). This equation states that real capital andbonds are assumed to be perfect substitutes (with equal net rates of return).Money market clearance (2.100) then implies that both the bond marketand the market for real capital are also cleared. This is so since excessdemand for these two assets togethermust be zero by the wealth constraint,and since capitalists are willing to accept any composition of bonds andreal capital in their portfolios.So much for stock equilibrium. Flow equilibrium for these three assets

now demands two conditions (see (2.102)). As already explained for thenon-bond situation, the government can always ‘‘sell’’ newly printedmoney and asset holders have been assumed to accept this inflow of moneyuntil they reallocate their portfolios in the ‘‘next period.’’ But can govern-ment also sell any desired amount of new bonds B� ? This is, in fact, assumedin (2.102), and can be rationalized by the perfect substitution assumption,whereby any arbitrarily small decrease in the return of bonds would induceasset holders to accept the new supply of bonds completely. Money andbond inflows are in this way assumed to be accepted by wealth owners (andare only subject to portfolio choices in the ‘‘next’’ instant). This nowpredetermines two nominal magnitudes in the savings decisions of wealthowners, and it assumes that the establishment of overall stock-marketequilibrium and of the full employment conditionK1 � n is brought aboutby the three equilibrating variables p,� and r — the price level, the rate ofprofit and the nominal rate of interest, respectively.The remainder of the model, including the implied kind of Say’s Law, is

then of the same type as in section 2.1 of this chapter and needs no furtherexplanation. This concludes the description of our bond-market extensionof the Tobin model.In order to derive again a system of autonomous differential equations

for this general equilibriummacrodynamicmodel (nowwith the additionalstate variable b), we have to consider its temporary equilibrium positionsfirst.In the present model, the (temporary) full employment condition reads

n�K1 � (S��M� /p�B� /p)/K (using (2.97))

� s�(�� rb� (m� b)�� t)� (m� b)� �

�m��b

(using (2.90), (2.92), and (2.102)),

where t is given by �(�� rb) and ��B1 �B� /B. This gives as first equationfor the temporary equilibrium variables �, m, and r.

n� s�(1� �)(�� rb)� (1� s

�)�(m� b)�

�m, (2.106)


since �b�B� /(pK)� (��

�)m by (2.99). The second equation is, as in

section 2.1, given by (using (2.100) and (2.101))

m� h�y� h

�(r�� r)(1� �), (2.107)

with

r�� /(1� �). (2.108)

We simply state here in advance that the unique steady state of this modelis given by

��

�� n, (2.109)

m�� h

�y, (2.110)

��

�m

�� n

s�(1� �)

, (2.111)

r��

��

�, (2.112)

b��

��

��

m�, (2.113)

which also provides us with the value of r�used in the above equilibrium

conditions. Of course, one has to check this assertion on the interior steadystate when the final dynamic equations have been determined.Inserting equations (2.107)—(2.108) into (2.106) gives

��n��b� (

�� (1� s

�)�)(h

�y� h

�(r��/(1� �) ))

s�(1� �)(1� b)� (

�� (1� s

�)�)h

�

, (2.114)

which shows the profit rate � as a function of � and b. Inserting this into theequation (2.107) implies dependence on � and b for this second equilibriumvariable. The temporary equilibrium part of the model therefore rests onquite lengthy expressions. This will change radically in the model of thenext section, where more stress is laid on lagged instead of instantaneousadjustment processes.In the followingwewill concentrate on the case s

�� 1, which gives us the

functions �(�, b),m(�, b) in the form

��n��b�

�(h

�y� h

�(r�� /(1� �)) )

(1� �)(1� b)� �h�

� �(�, b), (2.115)

m� h�y� h

�(r�� )(1� �)�m(�, b). (2.116)

Equation (2.115) implies �%� 0 near the steady state and also

�� (b��h�/(1� �))/((1� �)(1� b)�

�h�),


��1

(1� �)(1� �)b�

�h�

(1� �)(b� 1)��h�

� 0.

Form� andm%we obtain from (2.116) that near the steady statem%� 0 and

m�� h�(�� 1)(1� �),

� h��

(1� �)b��h�

(1� �)(b� 1)��h�

� 1� ,��

h�

(1� �)(b� 1)� �h�

� 0.

The dynamical equations for � and b, the state variables of the presentmodel, are given by

�� (��(1��)� ��)(�

� n��)� ��m,b� � (

��

�)m�

�b� bm.

These equations are obtained from (2.98) and (2.105) by noting thatp�

�� n� m holds by definition, i.e. p��

�� n�� m, and be-

cause (again by definition) of b1 �B1 � p� n� (��

�)m/b� (

�� m) by

use of (2.99). For m we have to insert into the above two laws of motionm� (m�/m)�� (m

%/m)b� , which gives:

�� (1��m�/m)� ��m%/mb� � (�� )(�� n��),

�� (�m�b/m)� (1�m%b/m)b� � (

��

�)m�

�b.

In matrix notation the last two equations may be expressed as

��

b� ��1� ��m�/m ��m%/m

�m�b/m 1�m%b/m�

��

�(�� )(�

� n��)

(��

�)m(�, b)�

�b� ,

�1

��1�m

%b/m ��m%/m

m�b/m 1� ��m�/m��(��)(�

� n� �)

(��

�)m(�, b)�

�b� , (2.117)

where

�� (1��m�/m)(1�m%b/m)��(m%/m) · (m�b/m)

� 1��m�/m�m%b/m.

This latter expression is zero for

�� m%b/m� 1

(m�/m),

It is now easy to check that the interior steady state is as described by equations(2.109)—(2.113).


which may be a negative or a positive number (or by chance 0). For all�� we know however that �� 0 must hold true (m�� 0,m

%� 0).

The Jacobian of the dynamical system (2.117) at its steady state is givenby

J�1

��1�m

%b�/m

��m%/m�

m�b�/m�1��m�/m��

�(��) 0

(��

�)m� (

��

�)m%�

�� .For �� we therefore get

det J��1

�(�� )((�

� �)m%�

�).

The expression for the det J is negative if

�(1�m

%)�

�, (m

%� 0),

holds true, which is in particular so if ��

�(b

�� 0) is assumed as a basic

reference case. We thus get for the general equilibrium Tobin model withbonds:

Proposition 2.9: Assume s�� 1,

��

�. The dynamic system

(2.117) for �, b is of saddlepoint type near the steady state.

In such a situation (and related cases) we thus have that the instabilityresult of section 2.1 for fast adaptive expectations carries over to the casewhere bond dynamics are included. This general equilibrium case is there-fore characterized, on the one hand, by low dimensional dynamics and onthe other hand, by a fairly contorted instantaneous feedback chain. Weshall see for the general Tobin model of the next section that the oppositewill hold for a system that favors disequilibrium adjustment processes inthe place of equilibrium conditions. The characterization given for theequilibrium model holds a fortiori when myopic perfect foresight ��

�(�� p) is assumed in the place of fast adaptive expectations. The result-ing dynamic system is then given by (for s

�� 1),

m� �� n� (1� �)��(m, b)� r

��m� h

�y

h�� ,

b� � (��

�)m� (m�

�)b,

since the LM equation

m� h�y� h

�(r�� (m, b)� p/(1� �)), p�

�� n� m,


then gives rise to a differential equations for the variable m. The function�(m, b) is obtained from the labor market condition (see (2.106) with s

�� 1)

n� (1� �)(�� (�� p/(1� �))b)��m,

where p has to be replaced by m and then by the above expression for m.There is a way by which the present general equilibrium version can be

completely reduced to the basic prototype considered in section 2.1 of thischapter. This possibility arises when the Barro definition of perceived dispos-able income of households is used in place of the concept so far employed(see Sargent 1987, pp.47ff., for details). This concept of perceived disposableincome assumes that government bonds do not constitute net wealth whenthe intertemporal government budget constraint is taken into account byhouseholds. In the context of the present macrodynamic framework thisleads to the following definition of Y�

�(see again Sargent 1987):

Y��

��K� rB/p� (M/p)��B� /p�T.

Together with the GBR G�T� rB/p�M� /p�B� /p, this gives

Y��

��K�M� /p� (M/p)��G��K� (

�� )M/p�G.

We assume now that g� �G/K is a constant (and ��

�), i.e., a fixed

(tax-independent) rule for government expenditures, but leave completelyopen, how much of the deficit G� rB/p�

�M/p(�T�B� /p) is paid

through taxes and how much through the issuing of new debt. The growthrate of the capital stock is in this case determined by (S

��

Y�� (1� s

�)Y��)

K1 ��S�� M�

p�B�

p��/K, (using the fourth equality in (2.92)).Recall that S

��Y��C�Y��L� (1� s

�)Y�

�(using (2.91) and

the first equality in (2.92)). From the second equality in (2.92) and (2.87) wethen have

S��K��L�

rB

p�T�C

��K�rB

p�T� (1� s

�)Y��.

Equating the two expressions for S�we obtain

Y�� K�rB

p�T��L.


The definition of disposable income under consideration is

Y��

��K�rB

p�M

p� �

B�

p�T,

hence

Y��Y��

� �L�M

p� �

B�

p.

Substituting into the first expression for S�we obtain

S��M

p��

B�

p� s

�Y��.

Hence

K1 ��M

p� �

M�

p� s

�Y�� /K��

M

p(��M1 )� s

�Y�� /K

��M

p(��

�)� s

�Y�� /K�m(��

�)� s

�

Y��K

�m(��)� s

�[� � (

��)m� g� ].

This is the growth equation (2.17) of section 2.1 for the capital stock in aneconomy without bonds. Coupled with the equilibrium conditionsn�K1 ,m� h

�y� h

�(r�� )(1� �), it therefore gives rise to the same

equilibrium relationships and the same dynamic equation for the statevariable � as in the basic version of the Tobin monetary growth model ofsection 2.1.We thus get that the real dynamics are independent of the bond dynam-

ics (not shown here), and also independent of how the ‘‘government deficit’’g� � rb�

�m is divided between taxes and new bonds. Ricardian equival-

ence makes the real economy independent of this composition of thegovernment financing structure. Of course, the volume of governmentexpenditure g� relative to the capital stock and the growth rate of themoneysupply will influence the real dynamics and its steady state.Despite its simplicity (only one law of motion), the model of section 2.1

therefore represents an important special case of general equilibriumgrowth dynamics with three assets and factor and product markets. Withthis extension of the equilibrium approach to monetary growth we leavethe realm of such models and will consider in the remainder of this bookthe alternative disequilibrium approach to monetary growth, structured ina specific hierarchical way, solely. For the development of the equilibrium


approach, which has dominated the literature until now, the reader isreferred to the survey article of Orphanides and Solow (1990) in particular.

2.5 A general disequilibrium version of the neoclassical model ofmonetary growth

We now are in a position to formulate a very general monetary growthmodel of the Tobin type by appropriately integrating the two precedingversions. This general version will mainly serve the purpose of representingthe proper point of departure for our subsequent formulations of inte-grated models of monetary growth of Keynes—Wicksell and of Keynesiantype, which both are models of labor-market as well as goods-marketdisequilibrium. Of course, one may always choose, if one prefers to do so,to return to general equilibrium supply-side versions of monetary growthdynamics as exemplified by themodel of the preceding section.This is themodel type that, due to its formal elegance and its logical consistency, butnot necessarily due to its relevance, has attracted the attention of mostmacroeconomists in the past two decades, as exemplified by the surveyarticle by Orphanides and Solow (1990). However our alternative ap-proach in the next chapter will be to dispense with the asset-marketdisequilibrium of this section, to acknowledge an independent investmentbehavior of firms and its financing by means of equities, and to shift theproblematic money-market disequilibrium description and its inflationaryconsequences to the market for goods, where price adjustment rules aremuch more plausibly established.

The equations of the model� are:�


��w/p, u��/x,� � (Y� �K��L)/K, (2.118)

W� (M�B)/p�K, p%� 1. (2.119)


Given by the double limit case ��, �

�� of the following version of Tobin’s

monetary growth model.� The alternative tax policy rule which holds (T� rB/p)/K constant (as in our later models

of Keynes—Wicksell and Keynesian type) in the place of the present formulation(T� (r��)B/p)/K� const. is not appropriate as a simplifying tool in the present Tobinmodel, where government bonds represent net wealth and are considered in the formationof perceived disposable income.

� The parameter � has to be removed from all equations of the following model if the secondalternative in equation (2.126) is chosen as the tax collection rule.


W� (M�B)/p�K,M� h�pY� h

�pK(1� �)(r� � r), (2.120)

Y��

��K� rB/p�M�B

p��T, (2.121)

C��L� (1� s�)Y��, s�

� 0, (2.122)

S��Y��C�Y� �K� rB/p�T�C� s

�Y��

�M�B

p�,

� (M� �B� )/p�K� , (2.123)

L1 � n� const. (2.124)


Y� yK(�Y�� y�K),L�Y/x[y,x� const.,V�L/L]. (2.125)

4 Government (fiscal and monetary authority):

T� �(�K� rB/p) [or t�� (T�(r��)B/p)/K� const.], (2.126)

G�T� rB/p��M/p [or (2.127)

� t��K�

�M/p], (2.128)

S��T� rB/p�G[�� (M� �B� )/p, see below], (2.129)

M1 ��, (2.130)

B� � pG� rB� pT�M� [� (��

�)M]. (2.131)


B�B : (1� �)r� �� (1� �)�, (2.132)

M� �M� ,B� �B� [K� �K� , see 6]. (2.133)

6 Equilibrium result (Say’s Law on the market for goods):

K� � S��S

��S�Y� �K�C�G�K� . (2.134)

7 Wage–price sector (adjustment equations):

w� ��(V�V� )�

�p� (1�

�)�, (2.135)

p��((M�M)/(pK))�

�w� (1�

�)�, (2.136)

�� (p��)��(�� n� �). (2.137)

Thismodelmerely represents the synthesis of themodels of sections 2.3 and2.4 and thus introduces no new relationships into our discussion of the


Tobin model of monetary growth. Of course, it also inherits the weak-nesses stated for its predecessors, which in our view can only be overcomeby extending this framework in the way we shall describe in chapters 3 and4.�Nevertheless, we have reached at this point a very general presentationof the Tobin approach to monetary growth which provides an importantstarting point for our future analysis of disequilibrium monetary growthdynamics. This very general Tobin model is indeed a very useful point ofdeparture for the further improvement of such general prototypemodels ofdisequilibrium growth toward the completeness, plausibility, and consist-ency of their building blocks and their assumed interactions.� We stresshere with respect to the government sector (2.126)—(2.131) that its mostbasic aspect is the formulation of its budget restraint. The above formula-tion of three simple policy rules by which this restriction can be filledrepresents only one of many examples of how to describe conceivablegovernment behavior within its budget restraint.As far as the mathematical investigation of this general Tobin model is

concerned, we will confine ourselves here to a presentation of its dynamicsin intensive form as well as an important special case, together with adetermination of its steady-state values and some general properties of itsJacobian. This general version of the Tobin model serves primarily thepurpose of preparing the groundwork for the Keynes—Wicksell model ofthe following chapter, which is there derived as a systematic modificationand improvement of this general disequilibrium version of a Tobin model.We shall close this section with some numerical simulations of this generalmodel of monetary growth.Let us first rewrite the general dynamical model (2.118)—(2.137) as a

five-dimensional autonomous dynamical system in the five variables��w/p, l�L/K,m�M/(pK),�, b�B/(pK). By calculations of the same

Note here that the following chapters will make use of the simple concept of actualdisposable income �K� rB/p�T in the place of the Hicksian one �K� rB/p� (M�B)�� /p�T. This means that the �-term in the formulation of the above taxcollection rule (and the government expenditure rule) will become superfluous and willsubsequently be removed from the model’s presentation.

� Note that the real wage and thus also the rate of profit is predetermined at each moment oftime — just as in our Tobin model with labor-market disequilibrium. This means that therate of interest is adjusted towards the rate of profit here, either because of the perfectsubstitutability assumption with respect to bonds and real capital holdings or simply bythe voluntary choice of the government. See Flaschel (1993) and Wolfstetter (1982) for asimilar treatment of the government sector in a Goodwin growth cycle model withoutmoney. Prices and wages finally are driven bymoney-market and labor-market disequilib-rium. This (problematic) scenario will be reformulated in the next chapter from theperspective of Keynes—Wicksell models.

� Since themagnitudes ofM� /p, B� /p and w,� are now predetermined in each moment of time,we here impose that S

�always fulfills the side condition S

��M� /p�B� /p� � �K.


type as in the preceding sections we obtain from (2.118)—(2.137) the inten-sive form equations:

�� [(1� �)��X�� (

�� 1)�

�X�], (2.138)

l1 � n� s( · ), (2.139)

m� �� n�� [�

�X��

��X�]� l1 , (2.140)

�� [��X�� X�]��(�

� n��), (2.141)

b� � (��

�)m� (�� n)b� ( (�

�X��

��X�)� l1 )b, (2.142)

where we employ the abbreviations:��

�� y� ��l, l� y/x� const.,X�� l/l�V� ,X��m� (h

�y� h

�(1� �)(r� � r)),

r�� /(1� �),t� �(�� rb),g� t� rb�

�m,

s( · )� s�(1� �)(� � rb)� (1� s

�)�(m� b)� (g� (t� rb)).

Note here that the above presentation of the dynamics again makes use ofthe relationship

p�� [��X��

��X�], � (1�

� �)�� (2.143)

for the deviation of the actual rate of inflation from the expected one (seesection 2.3) and that the s( · ) equation can be easily obtained from

s( · )�K1 �S�/K�S

�/K� y� ��C/K�G/K,

by use of the consumption function of the private sector and the govern-ment expenditure rule.We disregard the boundary steady state solutions �, l,m� 0 (caused by

the growth rate formulation of the corresponding laws of motion) in thefollowing determination of steady-state solutions of the above dynamics.These values of the variables�, l, and m are economically meaningless andwill not appear as relevant attractors in the stability investigations to beperformed. A general and global analysis of the system should of coursetake the stability properties of such steady-state boundary points of the

�� Note with respect to the s( · ) equation that the term (1� s�)�(m� b) is solely due to the

distinction drawn between the actual and the perceived disposable income of asset owners.This expression, and with it the Tobin effect, will disappear from the growth equation ofthe capital stock if perceived disposable income is set equal to actual disposable income, aswill be done in all following chapters of the book. Combined with the assumptiont�� t� rb� const., this provides one way in these later chapters of making the GBRirrelevant for the dynamics of the rest of the model.


dynamics (2.138)—(2.142) into account. For simplicity, we also assume herethat the parameter r� in the above model is always equal to the steady-statevalue r

�. This assumption simplifies the calculation of the steady-state

values without loss in generality, but it should be kept inmind or dispensedwith if steady state comparisons are being made.

Proposition 2.10: There is a unique steady-state solution or point ofrest of the dynamics (2.138)—(2.142) fulfilling �

�, l�,m

�� 0. This steady

state is determined by

l�� y/(xV� )[l

�� l

�V� � y/x� const.], (2.144)

m�� h

�y, (2.145)

��

�� n, (2.146)

b�� (

��

�)m

�/

�, (2.147)

��n�

�m

�� s

��b�� (1� s

�)�

�(m

�� b

�)

s�(1� �)(1� b

�)

,

r��

��

�/(1� �),

�� (y� ��

�)/l

�. (2.148)

Proof: Setting the right hand side of the equations (2.139) and(2.149) equal to zero implies that

�� n�� [�

�X��

��X�] must

hold in the steady state. Inserting this relationship into the right hand sideof (2.141) then gives that �

��

�� nmust hold in the steady state. This in

turn implies by (2.143) the equality of p and ��. From equations (2.138) and

(2.140) we then get the following system of equations for the variablesX�,X�

0� (1� �)��X�� (

�� 1)�

�X�,

0��X��

��X�.

It is easily shown for � �� 1 that this linear equation system can be

uniquely solved forX� andX�, and that these two terms in turn must bothbe zero. This implies the first two of our steady-state equations (2.144),(2.145). Equation (2.145) immediately follows from our assumption r� � r

�and (2.146) has already been shown above. Next, (2.147) is obtained bysolving the b� � 0 equation for the steady-state value of b (recallX�,X�� 0). The equation for �

�is then obtained from (2.139), i.e., n� s( · ),

by solving this equation for ��, since we have �

�� t�

��

(1� �)(1� b�)�

��

�b�and g

�� t�

��

�m

�in the steady state. The cal-


culation of ��, r

�is then straightforward. This concludes the proof of

existence and uniqueness of the interior steady state solution.�

We assume with respect to this steady-state solution first of all that theparameters of the model are chosen such that �

�� 0 holds true. This is

obviously the case if the growth rate of the money supply �and the

parameter �are set equal to the natural rate of growth n, since the tax

parameter � must satisfy � � (0, 1). The case just described can be regardedas the basic steady-state configuration of the general model, since thegovernment then just supplies the correct amount of money for the growthpath of the real part of the model and it injects this necessary amount ofnew money by buying goods (in addition to the ones that are financed bytaxes), i.e., there is no need for government debt or credit in this situation(b

�� 0!). The steady-state rate of profit is in this case simply given by

(n��m

�)/(s

�(1� �) )� 0,m

�� h

�y. Secondly, we must here also assume

that this expression for the rate of profit is less than y� �, so that there is apositive steady-state level of the real wage�

�that is associated with it. This

should always be fulfilled since the magnitudes of n, h�, and

�are all small

from an empirical point of view. On the basis of these assumptions we thushave a unique and meaningful interior solution to the steady-state equa-tions. It is assumed that the parameters of the model in general do notdepart by so much from those of this basic steady state configuration thatthe conditions �

�,�

�� 0 will be violated. Note, finally, that �

��

�� n

should not be chosen so negative that r�� 0 will not hold true. All

following investigations will be confined to stability considerations of oraround this steady-state solution of our models.Let us next consider the case of a tax collection rule of the form

t�� t� (r��)b� const. supplemented by g� t�

��

�m as the specifica-

tion of the government expenditure rule and show that these two assump-tion allow a reduction by one in the dimension of the dynamical systemunder consideration.�� We here, in addition, simplify the notation of theresulting four-dimensional dynamics by setting the value of V� equal to 1,allowing thereby for values of V larger than one.The variable b only enters the equations (2.138)—(2.141) via the s( · )

�� Note that we have to make use of t�� (T� rB/p)/K in the place of t��in the following

chapters, due to the different concept of perceived disposable income used in thosechapters, to allow for the same reduction in the dimension of the dynamical system underconsideration. The equation (2.150) is then to be replaced by

l1 � n� (s�(�� t�)�

�m),

and the excess demand situation X� that governs the movements of prices will then beredefined, as will be the nominal rate of interest r. These, however, are all changes that willbecome necessary in the following chapters on the Keynes—Wicksell and the Keynesianprototype model.


equation. In view of the above two respecifications of tax and expenditurepolicy, we get for s( · ) the expressions

s( · )� s�(�� (t� rb))� (1� s

�)�(m� b)� (g� (t� rb))

� s�(�� t�

�� b)� (1� s

�)�(m� b)� (g� t�

��b)

� s�(�� t�

�)� (1� s

�)�m� (g� t�

�)

� s�(�� t�

�)� (1� s

�)�m�

�m,

the last of which shows that the growth rate of the capital stock no longerdepends on the path of the stock of bonds, but only on the dynamicvariables �(�),m, and �. Furthermore, the entry J

��in the Jacobian J of the

dynamical system (2.138)—(2.142) is in this case simply given by �(�� n)

at the steady state of this system. The eigenvalue structure of the dynamicsat the steady state is therefore given by the eigenvalues (�

�� ) of the

four-dimensional system (2.149)—(2.152) below plus the eigenvalue�� (�

�� n). Stability assertions on the subsystem (2.149)—(2.152)

therefore immediately hold for the complete model (2.138)—(2.142), at leastfrom a local point of view.Thus, we consider the four-dimensional subsystem

�� [(1� �)��X�� (

�� 1)�

�X�], (2.149)

l1 � n� (s�(�� t�

�)� (1� s

�)�m�

�m), (2.150)

m� �� n�� [�

�X��

��X�]� l1 , (2.151)

�� [��X�� X�]��(�

� n��), (2.152)

where X�� l/l� 1, X��m� (h�y� h

�(r�� r)), �� y� � ��l and

r�� .Note again that y,x� const. and that y/x� l holds. Note furthermore

that the steady-state solution of the five-dimensional system is now to bebased on the expression

�� t�

��n�

�m

�� (1� s

�)�

�m

�s�

.

All other expressions for the steady state remain unchanged under theabove simplification of the dynamics.We note, finally, that the assumption t�� t� rb� const. will not be

sufficient here to suppress all influences of the dynamic variable b on therest of the system. This is due to the Tobin effect in the growth equation forthe capital stock. The dynamics of b, and thus the GBR, will, however,exercise no feedback on the remaining dynamical equations if, instead ofthe above, s

�� 1 is assumed (no Tobin effect). A third way of making the

GBR irrelevant is provided whenBarro’s definition of perceived disposable


income and t�� const., g� t��m is used (see the preceding section),

which transforms the K1 equation to the form K1 � s�(�� g)�

(1� s�)(

��)m, thereby again allowing a reduction by one in the dimen-

sion of the dynamical system to be studied.A further reduction in the dimension of the independent dynamical

system can then be obtained, as we know from the preceding sections, fromthe assumptions �� 0,�� . These assumptions imply ��

�� n,

which suppresses the Mundell effect, given by the positive dependence ofthe inflation rate on the expected rate of inflation,�� by keeping inflation-ary expectations constant along all trajectories of themodel. These reduceddynamical systems (with an appended b� equation) represent basic disequi-librium versions of neoclassical models of monetary growth to be com-pared with the corresponding models of the next two chapters.Let us finally once again consider the original Tobin scenario of govern-

ment policy in the framework of the extendedmodel. This scenario rests onthe assumptions B,G� 0,

��

�, which implies t��

�m. We then get

for K1 the differential equation K1 � y� �� c, which, by use of (see equa-tion (2.122)) c��l� (1� s

�)[�� m� t], becomes K1 � s

��

(1� s�)(

��)m, also in the case of disequilibrium in the market for

money and for labor.We do not derive here any stability results for the system (2.138)—(2.142),

but turn immediately to the dichotomizing model (2.149)—(2.152).

Lemma 2.1: Assume �� n, i.e. �

�� 0. Then, the determinant of

the Jacobian of the dynamical system (2.149)—(2.152) at the steady state isalways positive. Furthermore, the determinant of this system enlarged by(2.142), the bond dynamics, is always negative.

Proof: As above; see proposition 2.7.�

Proposition 2.11: Assume ��

�� n and

�sufficiently small.

Assume furthermore �� (�� ), i.e., ��

�� n. Then, the

steady state of the three-dimensional dynamics (2.149)—(2.151) is locallyasymptotically stable for all �

�, h


Proof: Analogous to the one in section 2.3 (also for s�� 1, as is

here allowed).�

Proposition 2.12:Assume the situation considered in the precedingproposition with ��. Then, the steady state of the four-dimensional

�� Here via the equation that determines the nominal rate of interest.


dynamical system (2.149)—(2.152) is locally asymptotically stable for all ��chosen sufficiently small.

Proof: As above, see proposition 2.8.�

Proposition 2.13:Assume ��

�� n and

�sufficiently small, h

�sufficiently large. Then there exists a minimal parameter value��

� 0(��

� 0) where the dynamical system (2.149)—(2.152) switches fromlocal asymptotic stability (up to this point) to local instability via a Hopfbifurcation as this value is passed through from below.

Proof: Calculating the trace of J at the steady state gives for theterms that involve either �

�or ��

��[(1�

�)h

�l�

��m

�� h�].

This term shows that �� will make the trace positive if the term in

brackets is positive, while �� will accomplish this in all situations. Notethat the remainder of the trace is given by�

�m

�� , and is assumed to

be negative. The application of the Hopf bifurcation theorem is then aroutine exercise; see Benhabib andMiyao (1981) for a comparable analysisof a system with a similar structure.�

The above are statements about the possible local behavior of the dynami-cal system (2.149)—(2.152). In order to go beyond these local results we haveto turn to numerical analysis, because the dimension of the dynamicalsystem and the interconnectedness of the equations make analytical ap-proaches very difficult, if not impossible.An example of such an investigation is presented below, but first, let us

briefly compare the components of the dynamical system (2.149)—(2.152)with those of the model of section 2.3 which, as we have just seen, providedpropositions of a very similar type. The basic difference in these twomodels(without and with a bond market) is the difference in the governmentexpenditure rule and the GBR. Thus we have t� g� �

�m in section 2.3,

and g� t��

�m in section 2.4.

Though the second model (of this section) has a bond market, itsinfluence on the remainder of the dynamics was suppressed, however, bythe assumption on t�

�. In this way, we can see that there is a close relation-

ship between the models of these two sections.By contrast, there is a huge difference between the models of this and the

preceding section. In the present section, instantaneous (static) relation-ships are trivial and do not create any problems for the calculation of thefive-dimensional dynamics based on them. In section 2.4, equilibrium


conditions are lengthy and the two-dimensional dynamical system basedon them is at first only implicitly given. Therefore, a variety of transform-ationswere needed before a single instability result could be derived. In thissection, significantly more dynamic laws are employed instead. Neverthe-less, due to the repetitive expression in the structure of these laws, analyti-cal propositions on these dynamics are possible and can surely be extendedbeyond those provided here.Chapters 3 and 4 will build on the general framework provided in these

last two sections by trying to remove certain obvious weaknesses step bystep. We shall therefore now always start from a five-dimensional dynami-cal system in general and only subsequently provide an analysis of some ofits important substructures. Furthermore, actual disposable income andperceived disposable income will be identical in our subsequent investiga-tions. Thus, in this book discussion of different concepts of disposableincome has been restricted to the framework of Tobin type models. Ofcourse incorporation of alternative definitions of disposable income in themodel types developed in later chapters remains an important futureresearch agenda.We close this section on the general Tobin model by some numerical

simulations of this model for the four-dimensional case just considered.The general parameter set for the following phase plots and time seriespresentations is shown in table 2.1. Parameter values will be taken fromthis set unless they are specified differently for specific simulation runs.The real (�Goodwin) growth cycle is obtained from the four-dimen-

sional model by setting �

� 1, �� 0 and ��

�� 0(�� 0). The

dynamic interaction between real wages � and the factor ratio l is at theheart of this cycle, which exhibits the closed orbit structure of the Goodwinmodel, as can be seen from figure 2.4. Note that the dynamical behavior ofreal balancesm does not feed back into the real part of this special economyand that inflationary expectations indeed remain stationary.Next, the pure monetary cycle is obtained from the four-dimensional

case by setting ��and

�equal to zero and by assuming l� l

�(l1 � 0). The

phase plots and time series displayed in figure 2.5 show that this monetarycycle is asymptotically stable and thus shrinking in amplitude, i.e., theCagan stability condition holds true for the chosen parameter set.One may now expect on the basis of the situations depicted in figures 2.4

and 2.5 that the interaction of these two cycle-generating mechanismsproduces again a convergent cyclical process. Yet, assuming

�� 0.5 as in

the general parameter set to allow for such an interaction adds the Rosemechanism to the real cycle (making the first entry in the Jacobian of thisdynamics positive, see section 2.3). Of course, in such a four-dimensionaldynamical system there may also exist other reasons that falsify the above


Table 2.1.

y�� 1, x� 2, l� 5.s�� 0.8, �� 0.1, n�

�� 0.05;

�� 0.1, t�

�� 0.08.

h�� 0.1, h

�� 0.06, i� 1, �

�� 1.

�� 0.05, �

�� 10, �� 1.3, �� 0,

��

�� 0.5.

Figure 2.4 Disentangled real cycle in the Tobin model

expectation, but surely the Rose type instability is the easiest to understandin this context. Be that as it may, the result of this coupling of two cycles isthe explosive real and monetary cycle shown in figure 2.6.In the present case, it can be shown that price flexibility works in favor of

economic stability and that wage flexibility does not. Therefore, assumingthe following nonlinear price adjustment function

��tan(c

�X�)/c

�(�� 10, c

�� 50)

in the place of our former choice ��X�,�

�� 10 should make the dynamical

behavior shown in figure 2.6 more viable. Note that this new adjustment


Figure 2.5 Disentangled monetary cycle in the Tobin model

function for the price level just states that price level flexibility will increasethe further the economy departs from goods-market equilibrium.� Thesimulation shown in figure 2.7 indicates that the working of this type ofnonlinearity leads to limit cycle behavior when the equilibrium is locallyunstable.Note that inflation rates are still very large in this last simulation of the

model and are thus not yet to be taken too seriously. Nevertheless, evenunder such extreme circumstances, the viability or boundedness of thedynamics is now guaranteed through the simple extrinsic nonlinearity inthe price level adjustment equation.

2.6 Outlook: independent investment behavior and Wicksellian pricedynamics

In this chapter on neoclassicalmonetary growth, we have assumed that netinvestment and the rate of capital accumulation are always fully deter-

� Where the extent of this disequilibrium must here fall in the range [�pi/100, pi/100], 0standing for goods market equilibrium and pi the real number �.


Figure 2.6 Combined real and monetary cycle of the Tobin model

mined by (i.e. identical to) the flow of savings and that the rate of change ofthe price level is determined by disequilibrium (or equilibrium) in themarket for money. These are surely questionable features of the Tobin typemodels of monetary growth.There are, in fact, many more drawbacks in this approach to monetary

growth, as will become apparent as our analysis proceeds, but one maynevertheless claim that these two features of the neoclassical approach areits most questionable ones. Goods-price dynamics is determined on themarket for goods (and not on the market for money) through equilibriumor disequilibrium conditions on this market. But in the present chapterthere exists no equilibrium condition (algebraic equation) or disequilib-rium condition (differential equation) with respect to the market for goodsfrom which the price level or its rate of change can be derived. Further-more, direct investment of savings is not the relevant characteristic of (atleast modern) market economies, which means that the basic forces thatdrive investment plans have to be determined and added to the generalmodel of monetary growth we have developed in this chapter.These fundamental weaknesses of the present macrodynamic model are


Figure 2.7 Combined real andmonetary cycle of the Tobin model with additionalnonlinearity in the price reaction function

overcome in the next chapter by (1) taking note of the fact that theinvestment decisions of firms are not identical to the savings decisions ofhouseholds, but are determined by variables that differ in an essential wayfrom the ones that determine the flow of savings, and (2) allowing fordisequilibrium on the market for goods, in the place of disequilibrium inthe market for money, in a very basic way through deviations between therate of investment and the rate of savings by making the rate of change ofthe price level now dependent on this discrepancy between the supply andthe demand for goods rather than for money.This modification of the Tobin or neoclassical type of approach gives

rise to the Keynes—Wicksell approach to monetary growth, the dynamicsof which was investigated with respect to some basic features from the latesixties up to the early eighties (see chapter 1).The following chapter extends and improves this type of dynamic analy-

sis considerably. Its main objective, however, is to show that there is asystematic improvement of the neoclassical approach to monetary growththat makes this monetary growth model more convincing from a descrip-


tive economic perspective, and that allows an extension to the dynamicresults obtained in the present chapter to a model with both labor- andgoods-market disequilibrium.There is thus some improvement and overhaul in certain modules of the

neoclassical model of monetary growth as we proceed to the next chapter.This puts this model on a firmer basis from a macroeconomic perspective,but is still far from representing the final improvement that has to be madein order to obtain a model of monetary growth that can be considered asthe working model for an analysis of Keynesian monetary growth.The alternative way of proceeding is, in view of the analysis presented in

this chapter, to return to the general equilibrium model of section 2.4 andto elaborate this type of analysis further. This is indeed the waymainstreammonetary growth theory has chosen to go, and that in particular into thedirection of micro founding the behavioral relationships that are employedthrough intertemporal optimizing behavior of households, firms, and thegovernment in a rational expectations environment with its typical saddle-path structure. In view of the numerous contributions that now exist in thisarea of research, the present book attempts to show that progress in thetheory of monetary growth may also come about in a complementary wayby laying stress on situations of disequilibrium in the market for labor aswell as the market for goods. This is done in the present book in a coherentframework which respects the budget restrictions of the various sectors,but which is not so decided with respect to the type of optimizing strategies(and their aggregation) that really take place within these budget restraintsand that may (or may not) lead to the aggregate behavior assumed in thesetypes of macromodels.


3 Keynes–Wicksell models ofmonetary growth: synthesizingKeynes into the Classics

In their book, Ferri and Greenberg (1989, ch. 4) present some nonlineardeterministic labor-market theories of business cycles which are based onthree variants of what they call the missing equation, i.e., the Phillips curvemechanism. The three approaches they consider are Rose (1967), Goodwin(1967) and Ito (1980), which make use of a special nonlinear money-wagePhillips curve, a linear real-wage Phillips curve, and a piecewise linear(regime switching) real-wage Phillips curve, respectively. In addition, anequation describing capital accumulation is derived in each of the threecases, and also a price adjustment equation in the case of Rose’s model.In the present chapter, we shall make use of a general model of Keynes—

Wicksell type� to show that these and other well-known models of cyclesand growth can all be considered as special cases of this prototype model,so that they all belong to one particular theory, which despite its ‘‘Keynes—Wicksell’’ origin is fairly (neo)Classical or supply-side oriented in nature.Such a statement does not, in our view, devalue this model type from aKeynesian perspective, but it leads us instead to a general and unifyingframework ofKeynes—Wicksellmodels�withwhichmodels that attempt tobe of a (more) Keynesian type can be usefully compared.Since our general Keynes—Wicksell prototype model synthesizes Good-

win’s Classical growth cycle and Rose’s ‘‘Keynesian employment cycle’’(based on sluggish wages and prices and smooth factor substitution), itmust inherit the dynamic features of these real models to some extent. This

� See alsoChiarella andFlaschel (1996a) for a discussion of thismodel type, wheremore stressis laid on a consideration of the government budget restraint and the occurrence of complexdynamics. � Which may also be called models of ‘‘supply side Keynesianism.’’

I.e., attempt to represent models of ‘‘demand side Keynesianism,’’ despite the presence of anelaborate wage—price module (representing ‘‘aggregate supply’’ as this is often called in theliterature).

127

result will, in fact, be shown in the following sections on the basis of a fixedproportions technology. Smooth factor substitution can be easily added toour model (see chapter 5), but we will not investigate here its (oftenobvious) implications. We here stress that such an extension does notintroduce a new theory of real wages into the model, since the marginalproductivity postulate does not represent a theory of real wages in thiscontext, as it is often incorrectly believed. Real wages changes are insteaddetermined by demand pressures on the market for labor and for goods,and they determine employment in a Classical fashion if smooth factorsubstitution is allowed for.The Classical nature of the model on this basis primarily arises from the

fact that output is determined through supply side conditions, i.e., we willhave full capacity growth throughout. The Keynesian IS—LM (dis)equilib-rium block here only serves to determine the rate of inflation, and it is fedback into the real part of the model via the real wage dynamics, expecta-tions, and the real rate of interest as one determinant of investmentbehavior. This is the Keynes—Wicksell portion of this predominantly(neo)Classical approach to monetary growth and cycles.In the next section, we present the general model. In section 3.2 we

present in intensive form the five laws of motion to which it gives rise. Wethen focus on the central four-dimensional subcase obtained when lumpsum taxes net of interest payments are held constant per unit of capital,which allows us to ignore the government budget constraint (GBR). Thisfour-dimensional subcase is the standard general reference for all of ourinvestigations of the considered prototype models. The remaining fifthdynamic law, the dynamics of the government budget constraint, is seldomexplicitly treated either in this chapter or in the subsequent chapters of thebook.It is our intention in this book to build the analysis of the most general

model on a systematic and detailed investigation of its important two-,three-, and four-dimensional subcases representing the private sector of theeconomy. This will already be very difficult in the four-dimensional case ofthis chapter, which therefore represents at present the final step in ourinvestigations of properties of the fundamental prototype models of mon-etary growth. From the viewpoint of completeness of such models, theGBR is nevertheless necessarily involved in their complete formulation andwill thus always be included in the initial presentation and explanation ofthe general case (here of the fifth dimension). In later work we intend to

Such an extension simply adds two further equations (�� f �(l), y� f (l)) to the model andtwo further unknowns (l,y), which in general leads to an increase in the stability of themodel.


study the role of the GBR and also various feedback policy rules that canbe built upon it in a systematic fashion.�Two-dimensional subcases of Goodwin and Rose growth cycle type

based on a number of simplifying assumptions are investigated in sections3.3 and 3.4 by means of Liapunov functions and the Poincare—Bendixsontheorem, respectively. One-dimensional discrete-time versions of thesemodels which can give rise to chaotic dynamics are also briefly considered,as well as Ito’s regime switching model which (in our reinterpretation of it)adds boundary conditions (ceilings) to the Goodwin growth cycle. Insection 3.5, interest rate flexibility is added to the Rose employment limitcycle via a less extreme formulation of money-market equilibrium, and isfound to imply that the limit cycle of the two-dimensional case disappears ifthe flexibility of the interest rate becomes sufficiently large. This section,however, still makes use of an extreme type of ‘‘asymptotically rationalexpectations’’ in its treatment of inflationary expectations, in order that theimplied dynamical system of dimension 3 remains. This allows for typicalapplications of the Routh—Hurwitz and the Hopf bifurcation theorems inthe characterization of the stability features and the cyclical properties ofthe system near the steady state.Sections 3.6 and 3.7, finally, consider various types of inflationary expec-

tations, the pure monetary cycle to which they can give rise and the generalfour-dimensional dynamics when this cycle is integrated with the real cycleconsidered previously. This four-dimensional case is investigated by meansof computer simulations and from the perspective of the various submodelswe have treated analytically in the preceding sections. Section 3.6 alsobriefly introduces the limit case model where product prices adjust with aninfinite speed, a case which has been very central in the literature onKeynesian dynamics and which will be taken up again at the end of chapter4 on the Keynesian prototype model.

3.1 The general prototype model

The following model type is derived by way of a systematic variation of thegeneral Tobin prototype model discussed at the end of chapter 2. Thesevariations concern the assumed investment behavior of firms and itsfinancing, asset-market equilibrium conditions, and the description ofgoods-market disequilibrium on which the theory of price inflation is nowbased. On the other hand, we now disregard for reasons of simplicity thefundamental distinction made in the Tobin models between actual andperceived disposable income of the household sector.

� See, for example, Chiarella et al. (1998) in this respect.

129Keynes–Wicksell models of monetary growth

The equations of the model are:�


��w/p, u��/x,� � (Y� �K��L)/K, (3.1)

W� (M�B� pE)/p, p

%� 1. (3.2)


W� (M�B� pE)/p,M� h

�pY� h

�pK(1� �)(r� � r), (3.3)

C� �L� (1� s�)[�K� rB/p�T], s

�� 0, (3.4)

S��L�Y�

��C�Y� �K� rB/p�T�C

� s�[ �K� rB/p�T ]� s

�Y�� (M� �B� � p

E� )/p, (3.5)

L1 � n� const. (3.6)

3 Firms (production units and investors):

Y� yK(�Y�� y�K),L�Y/x[y,x� const., V�L/L], (3.7)

I� i(�� (r� �))K� K[� n], (3.8)

pE� /p� I� (S� I)�S�S

��S

��Y� �K�C�G. (3.9)

K1 ��I/K� (1��

�)S/K

� I/K� (1� ��) (S/K� I/K),�

�� [0, 1], (3.10)

N� � ��K� �

�(S� I). (3.11)


T� �(�K� rB/p) [or t�� (T� rB/p)/K� const.], (3.12)

G�T� rB/p��M/p, (3.13)


M1 ��, (3.15)

B� � pG� rB� pT�M� [� (��

�)M]. (3.16)

� The parameter � has to be removed from all equations of the following model if the secondalternative in equation (3.12) is chosen as the tax collection rule (in which case �� (t�� rb)/(�� rb). Note here also that the money demand function which we employ in (all of ) thefollowing represents an appropriate linearization of its following general formM�M(pY,��, (1� �)r� �, pK), where the term pK is used for the time being as a proxy— for reasons of mathematical simplification — of the influence of nominal wealth pW onmoney demand (see Chiarella et al. 1998 for the inclusion of more general representations ofsuchwealth effects). Note, finally, that themagnitudeE� �E� in the followingmodel can alsobe negative — in the case in which the supply of new money and new bonds exceeds privatesavings. In this extreme case, firms sell so much from their inventories that they can financeinvestment from these ‘‘windfall profits.’’




�pK(1� �)(r� � r)[B�B,E�E], (3.17)

pE� (1� �)�pK/((1� �)r��), (3.18)

M� �M� ,B� �B� [E� �E� ]. (3.19)

6 Disequilibrium situation (goods market):

S�S��S

��Y� �K�C�G, (3.20)

I� i(�� r��)K� nK, (3.21)

S� I.


w� ��(V�V� )�

�p� (1�

�)�, (3.22)

p��((I�S)/K)�

�w� (1�

�)�, (3.23)

�� (p��)��(�� n� �). (3.24)

The important innovation of this general Keynes—Wicksell prototypemodel is the assumption that investment plans (now of firms) are doneindependently of the savings decisions of asset owners, up to the fact thatthey will be confronted and in some way or another be coordinated withthese saving plans through market interactions. This new fact, in conjunc-tion with the assumed LM-equation (3.17), can be viewed as being respon-sible for the label ‘‘Keynes’’ in the name of this model type. The particularform of the investment function (3.8) and the particular determination ofthe inflation rate (3.23) is responsible for the name ‘‘Wicksell’’ in this type ofliterature.The foregoing are the obvious and generally documented characteristics

of models of Keynes—Wicksell type, while further necessary consequencesof these changes in comparison to the model of the preceding chapter havebeen by and large ignored in the literature. This is in particular due to thefact there did not exist a general model of the Tobin type from which thisnew model could be obtained through systematic variations of its struc-tural equations, and with which the Keynes—Wicksell model could becompared in detail. Our following discussion of the new equations of thisKeynes—Wicksell model (in comparison to those of the model of section2.5) will indeed show that it represents a very systematic variation of thisformer model which improves it considerably with respect to plausibility,completeness, and consistency.


The newly added investment function (3.8) assumes that investment perunit of capital is determined in a natural, and here linear, way by thedifferential that is now allowed to exist between the rate of return � oncapital and the real rate of return r�� on government bonds. Thisdifferential was zero in the Tobin models of monetary growth in whichcapital (held by households) and bonds were perfect substitutes. There is afurther trend term in this investment function which is here for simplicityset equal to n (see chapter 7 for an endogenization of this term).Investment is assumed in this model to be entirely financed by equities

issued by firms. The asset structure that is available to capitalists (or pureasset holders) therefore now consists of outside money, government bonds,and equities (see Sargent 1987, p.12, for the same starting point). Equitiesand bonds are assumed to be perfect substitutes in the eyes of asset holders,which represents the most basic assumption that can be made in thiscontext. We assume in this model that there are no planned retainedearnings of firms, which means that all expected profits �pK are paid out toequity owners in each period. The after-tax return per unit of equity toequity owners is therefore (1� �)�pK/E. The price of equities (determinedby the above perfect substitute assumption) is denoted by p

(p%� 1 the

price of bonds). Thus the actual rate of return on equities per unit of moneyis given by (1� �)�pK/(p

E). Under the perfect substitute assumption this

must be equal to (1� �)r��, the real rate of interest after taxes, which isthe context of the equation (3.18).�Wealth owners now hold equities in place of real capital, which is under

the command of firms with regard to its use for production as well as withregard to its intended rate of change in time. Thus we have to replace thereal wealth componentK in asset owners’ portfolios by p

E/p, with respect

to actual holdings as well as with respect to stock demand (giving rise to a

Note here that the expected inflation rate � used in the calculation of the real rate of interestrepresents an average over themediumrun in our interpretationof thewage—price dynamicsof module 7 of the model (see also section 2.5). In principle, this also requires that theexpected rate of return and the nominal rate of interest are to be considered as representingsuch averages. This is easily done by assuming certain dynamic feedback rules for theseaverage concepts in view of their short-run equivalents (see Flaschel, Gong, and Semmler1998 and Chiarella et al. 1998). Due to the hierarchical order of the models developed andinvestigated in this book, such extensions are here left for the future in order to proceed in asystematicway from low to high dimensionalmacrodynamicmodel building. Furthermore,one might also argue on empirical grounds that investment depends negatively on expectedinflation �, as faster inflation may create an uncertain environment for investors. The rolethat inflationary expectations will play in the following is thereby reversed. We do not gointo this topic here any further, but will adhere to the traditional way in which theinvestment function has been formulated in models of Keynes—Wicksell type. The topicsindicated in this footnote must therefore be left for future research.

� Note that we here follow Sargent (1987, p.18) and assume that the expected change in theprice of equities is zero.


new form of Walras’ Law of Stocks). Furthermore, the savings decision ofcapitalist households now, of course, includes, besides money and bonds,the term p

E� /p, i.e., that part of private savings that is intended to go into

equities. These aspects are reflected in equations (3.2)—(3.6).Note here that we stick to the assumption that all taxes are paid by

capitalists. Note, furthermore, that we no longer distinguish between theactual and the perceived disposable income of capitalist households. Thisdistinction has been extensively treated in the preceding chapter, so that itmay be admissible to use the simple income concept Y� �K�T asperceived disposable income for the private sector as a whole from now on.Note here finally that this income is based on production plans and not onactual sales, just as in the models of Tobin type. Of course, the discussion ofmore elaborate concepts of perceived disposable income needs to be pur-sued in future investigations of the models proposed in this book.Firms issue equities in order to finance investment, and they have by

assumption no retained earnings with respect to their planned productionand planned proceeds. Investment may and will differ from total savings inmodels of Keynes—Wicksell type in general which means that plannedproduction and proceeds �pK and actual sales and proceeds will bedifferent from each other. The amount of production that is not sold isgiven by S� I�Y� �K�C� I�G.� Yet, this additional productionhas already (by assumption) been paid out to equity holders, which meansthat firms have to issue new equities as described in (3.9), not only in orderto finance their investment, but also to finance any difference betweenexpected and actual proceeds. Newly issued equities are therefore equal inamount to total savings, which implies that private savers will be justcontent with the supply of new equities by firms.Since we have independent investment behavior with I�S in general,

there is now the choice between investment goods supplied or demanded inthe determination of actual capital accumulationK� . These two polar casesare described in (3.10) by means of the parameter �

�(� 0, 1), and are to be

discussed briefly with respect to their consistency in the light of the otherequations of the model.Let us first consider the case �

�� 0, which is identical with the K�

assumption of the Tobin type models. In this case, we assume that firmsinvoluntarily invest their extra supply of goods in new machinery andfinance this extra investment as described above by issuing further equities,if supply Y exceeds aggregate demand C� I� �K�G. In the oppositecase, where I� S� 0 holds, they are forced to cancel this amount of theirinvestment plans and orders by assumption. In the present model the only

� The expression �K represents that part of production that is kept by firms for capitalreplacement purposes and for voluntary inventory changes.


consequence of these actions of firms is therefore given by the price adjust-ment equation (3.23), which says that any discrepancy between demandand supply C� I� �K�G�Y� I�S gives rise to correspondingprice movements according to the so-called law of demand. The immediateconsequences of goods-market disequilibrium are thus purely nominal inthe present model. In comparison to the model of section 2.5 this neverthe-less represents a significant improvement, since the price level is here nolonger driven by an imbalance in the market for the stock of money (animbalance which does not exist in the present model, see (3.17)), but in aWicksellian fashion by relative imbalances in the market for goods. Thepicture that emerges from this discussion of the case �

�� 0 is that of a

supply driven economy, as in section 2.5, but now one with a Keynes—Wicksell goods and money market demand block which determines therate of inflation p and the nominal rate of interest r. Since all goodsproduced are used for consumption or investment purposes in the presentcase there is no need to consider inventory changes N� explicitly. This isobtained from equation (3.11) by setting �

�� 0 in addition to the assump-

tion �� 0, i.e., equation (3.11) can be ignored in this case.

The latter remark is not true for the alternative case �� 1, where capital

accumulation is assumed to be driven by investment plans and not byintended savings. In this case there must be corresponding movements ininventories N which are determined by the imbalance in the market forgoods as described in equation (3.11). Inventories increase when outputexceeds aggregate demand (S� I) and they decrease in the opposite case(S� I). Note here also that we are considering a growing economy, whichmeans that there is a further reason for ongoing inventory changes, namely,that inventories have to grow in order to stay in line with the permanentgrowth in production and the capital stock. For simplicity, we assume herethat a certain portion of output (and thus of the capital stock) is retained byfirms for this purpose so that these inventory changes can be treated simplyas capital depreciation and simply be aggregated with it (��

��

�), just

representing a portion of actual production that (generally) does not leavethe sphere of production. The case �

�� 1 thus can be characterized as

being more demand oriented than the case �� 0 and thereby perhaps

somewhatmore in line withKeynesian concepts of monetary growth. Notethat equation (3.9) is also valid in this case, meaning again that firms haveto finance new investment and dividends that are not yet backed up bysales, but represented only by an increase in inventories (if S� I holds; inthe opposite case we instead have that part of the new investment isfinanced by unexpected sales from inventories). Again the immediate effectsof goods-market disequilibrium are purely nominal ones.The description of these two polar cases shows that intermediate cases


are also conceivable (�� (0, 1)) where any unsold production goes partly

into unplanned inventory changes and partly into unintended real capitalformation,�� with obvious changes in this description if investment de-mand exceeds currents savings.�� We shall, however, pay no attention tothis intermediate case in the following, but simply state here that itsstability properties will in fact be intermediate with respect to the ones weshall establish for the two polar cases.We thus end up here with a significantly revised description of the

behavior of firms (which induces only minor changes in the description ofhousehold behavior as we have seen above). By contrast there is no changenecessary in the formulation of the government sector when going from thegeneral Tobinmodel to this general version of a model of Keynes—Wickselltype.As already stated, we are no longer dependent here on money-market

disequilibrium in the formulation of an explicit (demand-pull) theory of therate of inflation p (see again (3.23), and note its use of a relative expressionfor the state of goods-market disequilibrium). Otherwise, the description ofthe wage—price module is the same as in the general Tobinmodel of section2.5. We thus have in this model the usual LM-equilibrium of Keynesianmodels which, by the wealth constraint of asset holders and the perfectsubstitute assumption for bonds and equities, implies that the other assetmarkets must be cleared as well (see (3.17)). The perfect substitute assump-tion (3.18) has already been explained above, while (3.19) again states thatasset holders will voluntarily accept the additional supply of money andbonds and adjust their resulting changed portfolios only in the ‘‘subsequentperiod’’. Due to the implied equality S�S

�� S

�� p

E� /p (see (3.5) and

(3.14)), we obtain from (3.9) the equation E� �E� (see (3.19)), i.e., generalconsistency with respect to flows�� (besides the general consistency forstocks (3.17)).This concludes our description of the general Keynes—Wicksell model of

this section (which, besides labor market disequilibrium, now also exhibits

�� Both being financed by issuing new equities, since firms have no earnings from currentproduction.

�� Note that this also covers the case of an excessive new bond and money supply(M� �B� � pS

�) by the government, in which case we can have negative aggregate savings S

and a reduction in the stock of equities financed by excessive sales of firms from inventories.A constraint of the type pS� p

E� �0 is therefore not really necessary in the present

formulation of the model. Note also that in such a case it is not only investment demand,but also other demand (here implicitly assumed to be satisfied before investment demand isconsidered), that is (completely) met by appropriate inventory changes. Also in such asituation it is therefore only investment demand that can be rationed in the present model.

�� Note here again that this ‘‘simple’’ assumption bears strong consequences with respect tothe ability of the government to influence the pace of capital accumulation. Nevertheless,we shall not dispense with this standard assumption of continuous-time macrodynamictheory in this book, but shall leave its detailed reconsideration for future investigations.


goods-market disequilibrium as the explanation of price inflation). Westress once again that it is mainly the firms sector which has received anextensive reformulation here accompanied by a new arrangement of equi-librium and disequilibrium conditions and their implication for the formu-lation of the wage—price module. In our view this model type is much moreconvincing than the general Tobin model of the preceding chapter. Never-theless, in discussing this model we shall find that it is still fairly neoclassi-cal in its structure and its implications due to some definite weaknesses itcontains. These weaknesses concern the description of goods-market dis-equilibrium and the treatment of unplanned inventory changes. In mosttreatments these weaknesses are generally simply removed from view bythe assumption of an infinite adjustment speed of prices, as we discuss in alater section of this chapter. Better ways to overcome these weaknesses(and the differences that this implies for the working of such a model) willbe the theme of the next chapter.As far as themathematical investigation of this general Keynes—Wicksell

model is concerned, we will confine ourselves here mainly to the caset�� t� rb� const., where lump sum taxes are varied in such a way thatthe ratio of taxes net of interest to the value of the capital stock remainsconstant over time. This assumption will allow us to disregard the GBRand the evolution of government debt in the following, at least from a localpoint of view. In making use of this simplifying device we here follow asimilar assumption of Sargent’s (1987, ch. 5) ‘‘dynamic analysis of a Key-nesian model,’’ which is the basic reference with respect to the models weshall investigate in this chapter and in chapters 4 and 5.

3.2 The intensive form of the model

Before we now start with the step-by-step investigation of the four-dimen-sional case with t�� t� rb� const., let us first rewrite the general dynami-cal model (3.1)—(3.24), without any simplifying assumptions, as an auton-omous dynamical system in the five variables � �w/p, l�L/K, m�M/(pK), �, and b�B/(pK).�

� Note that the system in fact exhibits two further laws of motion for the variables ��N/Kand e�E/K which, however, do not feed back to the other laws of motion of the model.These two laws read:

��

�(s( · )� i( · )� n)� n�� l1� [�

��

�/n],

e�(1� �)r� �

(1� �)�s( · )� n� l1 [e

�indeterminate].

It is, of course, necessary to check that both � and e remain nonnegative and finite in thecourse of the dynamic evolution of the above dynamics. Note that the second law impliesthat the number of equities grows with the rate n in the steady state, while bonds B andmoney M both grow with the rate .


By calculations of the same type as in the preceding chapter (see inparticular section 2.3) we obtain from (3.1)—(3.24) the intensive form equa-tions:

�� [(1� �)��X�� (

�� 1)�

�X�], (3.25)

l1 � n� s( · ) or �i(·) (�� 0 or 1), (3.26)

m� �� n�� [�

�X��

��X�]� l1 , (3.27)

�� [��X�� X�]��(�

� n��), (3.28)

b�� (

��

�)m� (�� n)b� ( (�

�X��

��X�)� l1 )b, (3.29)

where we employ the abbreviations:

�� y� ��, l�L/K� y/x�const.,X�� l/l�V� � y/(xl)�V� ,X�� i( · )� n� s( · ),r� r� � (h

�y�m)/(h

�(1� �))[h(y, r)� h

�y� h

�(1� �)(r� � r),

see (3.3)],t�T/K� �(� � rb), t�� t� rb,g� t��

�m,

s( · )� s�(�� t�)� (g� t�),

i( · )� i(�� r��).

Note here that in the above presentation of the dynamicswe havemade useof the formula p�� [�

�X��

��X�] for the deviation of the actual

rate of inflation from the expected one, and that the s( · ) equation can beeasily obtained from s( · )�K1 � y� ��C/K�G/K by inserting into itthe consumption function and the government expenditure rule.In the following determination of steady-state solutions of the above

dynamics we again disregard the boundary solutions �, l,m� 0 whicharise from the growth rate formulation of certain laws of motion. Thesevalues of the variables �, l, and m are economically meaningless and willnot appear as relevant attractors in the stability investigations to beperformed. A general and global analysis of the system should, of course,take into account the stability properties of such boundary points of rest ofthe dynamics (3.25)—(3.29). For simplicity, we also assume here that theparameter r� in the above model is equal to the steady-state value r

�. This

assumption simplifies the calculation of the steady-state values withoutloss in generality, but it should be kept in mind or dispensed with if steadystate comparisons are being made.


Proposition 3.1: There is a unique steady-state solution or point ofrest of the dynamics (3.25)—(3.29) fulfilling �

�, l�,m

�� 0.� This steady

state is determined by:��

y�� y�, (3.30)

l�� y

�/(xV� ), l

�� y

�/x, (3.31)

m�� h

�y�, (3.32)

��

�� n, (3.33)

b�� (

��

�)m

�/

�, (3.34)

��n�

�m

�� s

��b�

s�(1� �)(1� b

�),

r��

��

�,

�� (y

��

�)/l

�. (3.35)

Proof:The equations (3.26) and (3.27) (set equal to zero) imply that�� n�� [�

�X��

��X�] must hold in the steady state. Inserting

this into (3.28) then gives that ��

�� n must hold. This in turn implies

by (3.27) the equality of p and ��. From the equations (3.25) and (3.27) we

then obtain the following simultaneous equation system for the variablesX� and X�:

0� (1� �)��X�� (

�� 1)�

�X�,

0��X��

��X�.

It is easily shown for � �� 1 that this linear equation system can be

uniquely solved forX� andX�, which must then both be zero. This impliesthe first two of our steady-state equations (3.30) and (3.32). Equation (3.32)then immediately follows from our assumption r� � r

�and (3.33) has al-

ready been shown above. Next, (3.34) is obtained by solving the b� � 0equation for the steady-state value of b (X�,X�� 0). The equation for �

�is

then obtained from (3.26), i.e., n� s( · ), by solving this equation for ��, since

we have �� t�

�� (1� �)(1� b

�)�

��

�b�and g

�� t�

��

�m

�in the

steady state. The calculation of ��, r

�is then straightforward (i( · )� 0!).

This concludes the proof of existence and uniqueness for the interior steadystate solution.�

� The following presentation of this steady state of the dynamics immediately implies thatmoney is not superneutral in this model, i.e., the rate of growth of the money supplyexercises an influence on the real side of the steady state of the model.

�� It is easy to calculate for the additional dynamic variablesN/K, pE/(pK) the steady-state

values: ��/n and (1� �)�

�/((1� �)r

��

�), respectively.


We assume with respect to this steady-state solution first of all that theparameters of the model are chosen such that �

�� 0 holds true. This is

obviously the case if the growth rate of the money supply �and the

parameter �are set equal to the natural rate of growth n, since the tax

parameter � must satisfy � � (0, 1). The case just described can be regardedas the basic steady-state configuration of the general model, since thegovernment then just supplies the correct monetary frame for the growthpath of the real part of the model and it injects this necessary amount ofnew money by buying goods (in addition to the ones that are financed bytaxes), i.e. there is no need for government debt or credit in this situation(b

�� 0!). The steady-state rate of profit is in this case simply given by

(n��m

�)/(s

�(1� �))� 0,m

�� h

�y. Secondly, we must here also assume

that this expression for the rate of profit is less than y� � so that there isassociated with it a positive steady-state level of the real wage �

�. This

condition should always be fulfilled since the magnitudes of n, h�, and

�are all small from an empirical point of view. On the basis of theseassumptions we thus have a unique andmeaningful interior solution to thesteady-state equations. It is assumed that the parameters of the model ingeneral do not depart by so much from those of this basic steady-stateconfiguration that the conditions �

�,�

�� 0 will be violated. Note, finally,

that ��

�� n should not be chosen so negative that r

�� 0 will not hold

true.Let us now start with the investigation of the case tn � t � rb � const.

We here in addition simplify the notation of the four-dimensional dynami-cs that results in this case by setting the value of V� equal to 1. Since thevariable b only enters the equations (3.25)—(3.28) via the s( · ) equation(which only depends on t�), we immediately see that the first four dynamicallaws and their components do not depend on the variable b. Furthermore,the entry J

��in the Jacobian J of the dynamics (3.25)—(3.29) is in this case

simply given by�(�� n) at the steady state of this system. The eigenvalue

structure (��

) of the dynamics at the steady state is therefore given bythat of the system shown below plus the eigenvalue �

�� (�

�� n).

Stability assertions on the subsystem (3.25)—(3.28) therefore immediatelyalso hold for the complete model (3.25)—(3.29), at least from a local point ofview.In light of the foregoing discussion we are led to consider the four-

dimensional system:

�� [(1� �)��(y/(xl)� 1)� (

�� 1)�

�(i(·)

� n� s( · ))], (3.36)

l1 � n� s( · ) or �i( · ), (�� 0 or 1), (3.37)


m� �� n� [�

�(i( · )

� n� s( · ))� ��(y/(xl)� 1)]� l1 , (3.38)

�� [��(i( · )� n� s( · ))� ��(y/(xl)� 1)]

��(�� n��), (3.39)

where

s( · )� s�(y� � ��y/x� t�)�

�m,

i( · )� i(y� � ��y/x� (r�� (h

�y�m)/h

�)� �).

Note here again that y,x� const. still holds (as in the last chapter) in thecontext of the Keynes—Wicksell model of this chapter and that the par-ameter � is now no longer present in this model variant. Note, furthermore,that the steady-state solution of the four-dimensional system is now basedon the expression �

�� (n�

�m

�)/s�� t�. All other expressions for the

steady state remain unchanged under the above modification.��The abovedynamic system will now be investigated by starting from an appropriatetwo-dimensional subcase of it.

3.3 The Goodwin growth cycle case

This section starts from a set of simplifying assumptions which imply thatthe real part of the Keynes—Wicksell model of this chapter gives rise todynamics of the Goodwin (1967) growth cycle type. The overshootingprofit squeeze mechanism of that model is thus an integral part of ourgeneral Keynes—Wicksell model.� In order to obtain the simple two-dimensional center type dynamics of

this growth cycle model from the above four-dimensionalmodel one has tomake the following four assumptions:

�� 1: The real wage dynamics is independent of the goods market.

r� r�: Infinite interest elasticity of money demand at the steady state

(h��.)

�� n: Extreme asymptotically rational expectations (��

�,��).

�� I.e., we have (U� ,V� � 1):

y�� y�, l

�� y

�/x, l

�� l

�,

��

�� n,m

�� h

�y�,

�� (y

��

�)/l

�, r

��

��

�� n.

Note here that Sargent (1987, ch. 5) obtains the superneutrality of this steady state byassuming g� const. and

�� 0 in addition to the above assumption t�� const.

� See Flaschel (1984, 1993), Flaschel and Sethi (1996), and Flaschel, Franke, and Semmler(1997) for various representations and investigations of the Goodwin growth cycle modeland its extensions, and Flaschel and Groh (1995) for some empirical observations on thismodel type that extend Solow’s (1990) reappraisal of this very fundamental model ofcyclical growth.


�� : Government is a creditor in the steady state: b

��m

�.

In the case K � i( · )� n (i.e. �� 1), the first three of the above assump-

tions are in fact already sufficient to imply the cross-dual growth cycledynamics of the Goodwin model for the real part of the model (3.1)—(3.24),since we then get from equations (3.36) and (3.37) the following specialdynamic equations (y,x� const., l� y/x):

�� (l/l� 1), (3.40)

l1 �� i(y� � ��l� r��

�� n). (3.41)

It is obvious from these equations that r� r�removes the influence of the

money market on the real part of the model (r��

��

�� n), that

�� n removes the dynamics of expectations formation, and that

�� 1 suppresses the impact of the goods-market disequilibrium on the

dynamics of the real-wage.Since l and y are given magnitudes in the model (3.1)—(3.24), the above

two equations are easily reformulated in terms of Goodwin’s originaldynamic variables u� �l/y��/x (the share of wages) and V� l/l (therate of employment):

u��(V� 1)� h�(V ), (3.42)

V1 � i(y� �� uy� r��

�� n)� h�(u). (3.43)

Proposition 3.2: The trajectories of the dynamical system (3.42)—(3.43) stay positive if they start in the positive domain of R� and are allclosed orbits.

Proof: It is easily shown that all orbits which start in the positiveorthant must stay in it, since the boundary of this domain is an invariantsubset of the above dynamics. The proof that all trajectories of this dynami-cal system are closed orbits is also straightforward if one makes use of thefollowing function

H(u,V )��"

"�

(h�(u& )/u& )du& ��'

'�

(h�(V� )/(V� )dV� .

This function is zero at the steady-state values u�,V

�and positive else-

where. Furthermore, one easily gets H� �H"· u� �H

'·V� �

(� h�(u))u� h�(V )V1 � 0, i.e., the function H is a Liapunov function.��Due to the shape of this function, it follows that all orbits must be closed(see Flaschel 1993, ch. 4, for the details of such reasoning). The resultingphase portrait of this dynamical system is well-known (see again Flaschel1993, ch. 4, for the graphical details).�

�� See Hirsch and Smale (1974, pp.192ff.), and Brock and Malliaris (1989, pp.94 ff.).


All observations in the preceding proof can be reformulated in a straight-forward way for the original presentation of the dynamical system (3.40)—(3.41) in the variables � and l, and they also hold for all nonlinear labormarket reaction functions �

�(l/l) with �

�(0)� 1, ��

�� 0 (see the next sec-

tion for the introduction of such nonlinear Phillips curves).The remaining dynamical equations of this growth cycle case are

m� � (��(i( · )� n� s( · ))�

��(V� 1))� i( · )� f �(u,V ),

b� �� m� (p( · )�K1 ( · ))b� f �(u,V,m, b).

Since we only want to show here thatGoodwin’s growth cycle is part of thefully interdependent dynamics of the general model, we do not discuss thisappended dynamical system in the special case we are considering in thepresent section. Of course, Goodwin’s type of dynamics will also be presentand tend to dominate if r� r

�(high interest elasticity of money demand)

and �

� 1, �� holds, but may be modified significantly in its over-shooting feature when less extreme parameter values are given.The Goodwin model is even more closely mirrored if the alternative case

K � s( · ) (i.e., �� 0) is considered. In this case, the further above assump-

tion �� 0 on government behavior is needed, if one wants the dynamics

of�, l (i.e., u,V) to be fully independent of the rest of the system. This is duehere to the form of the savings per unit of capital function

s( · )� s�(y� � ��l� t�)�

�m.

The belief that (real) wage flexibility will give rise to full employment steadygrowth, at least in the long-run, is supported most when Kuh’s (1967)version of the Phillips curve is used in the Goodwin context, see Akerlofand Stiglitz (1969, pp.272—274) for such an application. This version of thePhillips curve can be formulated as follows (see Ferri and Greenberg 1989,p.75). Set

�� (V )y/l, i.e., u� �

�(V ).��

With respect to the model (3.42)—(3.43), this latter equation replaces(3.42) and gives, in conjunction with (3.43):

V1 � i(y� ��(V )y� r

��

�� n) or

V� � iy(��(V

�)� �

�(V ))V�H(V ),

where V�is defined by �

�(V

�)� 1� (�

�� )/y,�

�� r

�� (

�� n) (u

��

��(V

�)). These values characterize the steady state of the model and we

assume here that an economically meaningful solution V�� 0 exists. This

steady state is obviously globally asymptotically stable, since we have

�� ( · ) may be nonlinear and is assumed to fulfill �

�(V ) � (0, 1),��

�� 0. Note that the

expression x� y/l in this new Phillips curve represents labor productivity Y/L.


V� � 0 to the left of V�andV� � 0 to its right. Employment decreases to the

right of V�and with it the real wage until income redistribution induces a

growth rate of the capital stock that is equal to the growth rate of the laborforce n (the opposite occurs to the left ofV

�). It has become common usage

to call 1�V�the natural rate of unemployment and to consider V

�as the

‘‘full’’ employment rate (see Akerlof and Stiglitz 1969, p.271, for an earlyexample of this). The model therefore gives the most straightforwarddemonstration of the long-run stability of the full-employment situation.Note, however, that the present explanation of ‘‘natural’’ employment

V��

�((y� � � r

��

�� n)/y) (assumed to lie between 0 and 1) is far

from being ‘‘natural,’’ as its dependence in particular on ��and

�shows.

Furthermore, even this simplemodel of growth and (un)employment cangive rise to complex dynamics if it is reformulated in discrete time, even if��(V ) is assumed to be a linear function of V. In this latter case it gives rise

to the following well-known difference equation that allows for ‘‘chaos’’ atappropriate parameter values for i, y and �

�, namely, V

!(��

V!(1� iy�

�(V

��V

!)) (see Pohjola 1981 and Ferri and Greenberg 1989 for

its treatment in this context). We thus can associate even chaotic behaviorwith this most basic form of a full-employment ‘‘adjustment’’ mechanism ifthe parameter �

�becomes sufficiently large (V

�sufficiently small) (see

again Pohjola 1981 for details).Ferri and Greenberg (1989) consider in chapter 4.8 another approach to

labor market dynamics which they call a neoclassical disequilibrium ap-proach. This approach, which is based on neo-Keynesian regime switchingmethods, takes account of the fact that the employment rate V cannotincrease beyond 1 if 1 stands for the ceiling of absolute full employment.��The Goodwin model (3.40)—(3.41), for example, has then to be modified asfollows (l a given magnitude):

�� (l/l� 1), (3.44)

l1 ��i(xl� � ��l� r

��

�� n),

�i(xl� ��l� r��

�� n),

if l� l

if l� l,(3.45)

to take account of the fact that employment and production cannotincrease beyond the full employment level: l� l, y� xl. This model isconsidered in Ito (1980) in full detail and with regard to the mathematicalcomplexities to which such a regime switching approach can give rise.There is, however, one fundamental shortcoming of such regime switch-

ing approaches which lies in the fact that they usually identify the steady-

�� Note here that we have used ‘‘1’’ in the four-dimensional dynamics (3.36)—(3.39) to denotethat level of employment where there is no money wage drift from the side of the labormarket, which is assumed to be a magnitude significantly below full employment.


state rate of employment with the maximum rate of employment. Such aview is not shared by many macroeconomists, quite independently of theparticular justification they may give for the assumption (or derivation) ofa positive magnitude V

�or 1�V

�, often called the natural rate of

(un)employment (or the NAIRU if a broader definition is given to thispositive steady state concept of (un)employment). We here use the value 1for V

�for simplicity to denote the ‘‘natural’’ level of the employment rate

and thus have to use V��

� 1 if we want to refer to some sort of absolutefull employment ceiling.Introducing such a full employment ceiling into the equations (3.40)—

(3.41) gives, instead of (3.44)—(3.45), the equations (l� y/x� const.):

�� (l/l� 1), (3.46)

l1 ��i(xl� � ��l� r

��

�� n),

�i(xlV��

� ��lV��

� r��

�� n),

if l� l/V��,

if l� l/V��.(3.47)

This implies that the dynamics are of the same type as those of (3.40)—(3.41)as long as l stays within (l/V

��,��), i.e., within a certain neighborhood

of the steady-state value l�� l � (l/V

��,��). Only if l falls below l/V

��is there such a shortage of the labor supply that output must fall below thepotential outputY�� yK and will thus modify the path of capital accumu-lation (and that of l). Of course, the Phillips curve may have kinks inaddition, as in Ferri and Greenberg (1989, p.62), at various levels of theemployment rate. This, however, only modifies the shape of the closedorbits of the Goodwin model, not its qualitative features.The phase portrait shown in figure 3.1 summarizes the above findings on

labor supply bottlenecks in theGoodwinmodel. Leaving aside such bottle-necks from the side of labor supply and (by the use of equations (3.11) and(3.23)), and also certain bottlenecks from the side of capacity outputincluding inventories, therefore simply means that the dynamics of system(3.1)—(3.24) is restricted to such a domain of economically meaningfulvalues where neither productive capacity plus inventories nor naturalcapacity (L

��L ·V

��) becomes a binding constraint for the growth

path K1 � i( · ) of the economy.�� Important as such switches in economicregimes may be from a global point of view, they can at first be safelyneglected in the study of the fundamental properties of the dynamic system(3.1)—(3.24) and its special cases. Ceilings to economic activity (caused bythe existing supply of goods including inventories)�� and the present

�� Which would lead to regimes of absolute goods supply shortages or absolute labor supplyshortages.

�� Note here however that equations (3.11) and (3.23) are not without problems, problems


Figure 3.1 Ceilings to the validity of the Goodwin growth cycle approach

volume of maximum labor supply are of no importance for the economicevolution near the steady state.Of course, they have to be added eventually to any global treatment of

the Keynes—Wicksell model, but will then not give rise to a new theory oflabormarket dynamics. Instead, there is only a switch in the determinationof the isocline l� � 0 below l/V

��as shown in figure 3.1 by the curveAB,�

given by

�� (x� (�� r�� (

�� n)))/(lV

��),

which should lead to it having a positive slope with respect to empiricallyplausible values of the parameters �, r

�, n and

�. This is, of course, no

significant modification of the dynamics of the Goodwin model.This last statement can be further substantiated by means of the

Liapunov function

which will be discussed and removed in chapter 4 when going from the Keynes—Wicksellprototype model to the Keynesian one.

� See also Akerlof and Stiglitz (1969, p.272) in this regard.


H(�, l)��

��

h�(�& )�&

d�& ��

��

h�(i� )

l�dl� ,

for the dynamical system (3.46)—(3.47) where h�(l) is given by ��(l/l� 1)

and h�(�) by�i(xl� ��l� r��

�� n). This function is of the same

type as the Liapunov function we used before and it gives rise to�

H� ��0 if l� l/V

��h�(l)[h�(�)� i(xlV

�� lV

�� r

��

�� n)] if l� l/V

��

� h�(l)[h�(�)� h�(�)]� 0.

This implies that the original closed orbits of the Goodwin model arecrossed inwards by the trajectories of this new dynamical system in theregion below l/V

��(see figure 3.1), so that the closed orbit of figure 3.1

that runs through A becomes a limit cycle for all trajectories that start atpoints outside of it. The closed orbits of the Goodwin model thus charac-terize this dynamical system in the long run also in the cases where regimeswitching takes place.

3.4 The Rose employment cycle extension

In this section we remove one of the simplifying assumptions of thepreceding section. We show that the limit cycle result of Rose (1967) canthen be obtained through the interaction of the Goodwin profit squeezemechanism (of the preceding section) and locally destabilizing and globallystabilizing relative adjustment speeds of wages and prices. These latterforces were the basic ingredients of Rose’s nonlinear theory of the employ-ment cycle.��We have considered in the preceding section four variants of Goodwin’s

growth cycle model and have argued in particular that it is far fromobvious that the real wage mechanism ��

�(l/l� 1), or even a simplifi-

cation of it, will guarantee full employment equilibrium in the long run.Smooth factor substitution with a sufficiently high elasticity of factorsubstitutionmay alter this conclusion to some extent, but only insofar as itthereby becomes an empirical question of whether Goodwin or Solowprovides the more convincing approach to the supply-side determinedpath of capital accumulation.In the present section we shall now demonstrate that the Solovian

� If � �x�Y/L, i.e. as long as profits remain positive.�� See Flaschel (1993), Flaschel and Sethi (1996), and Flaschel, Franke, and Semmler (1997)

for various representations and investigations of the Rose employment cyclemodel and itsextensions.


outcome (of a monotonic convergence to the full-employment growthpath) becomes even more unlikely if it is realized, as in Rose’s (1967) modelof the employment cycle, that even in a supply-side driven economy theevolution of real wages is driven not only by the disequilibrium in the labormarket but also by disequilibrium in the market for goods. This proposi-tion also extends to the case of smooth factor substitution as Rose (1967)has already shown with a similar real growth model. The essential ideasbehind his employment limit cycle are, however, also more easily graspedin the context of a fixed proportions technology as we shall show in thissection.In order to obtain a Rose type model as a special case of our general

framework (3.1)—(3.24) we have only to assume �� 1�� as modification of

the assumptions of section 3.3 (all other assumptions of that section remainintact). The Goodwinian dynamical system (3.40)—(3.41) is thereby extend-ed to the dynamical system (see also equations (3.22) and (3.23)):

�� [(1� �)��(l/l� 1)� (

�� 1)�

�(i( · )� n� s( · ))], (3.48)

l1 �� i( · ), (or n� s( · )), (3.49)

where i( · )� i(y� � ��l� r��

�� n) and s( · )� s

�(y� ��l� t�),

as in section 3.3.In order to study the dynamics of this extended model, let us again

consider the caseK� � I (i.e. �� 1) first.Making use again of the Liapunov

function

H(�, l)��

��

h�(�& )�&

d�& � ��

��

h�(l� )

l�dl� ,

where h�(l)� (1� �)��(l/l� 1), h�� i(�). This Liapunov function is

of the type we have considered for the system (3.40)—(3.41) in the precedingsection. Here we obtain the following proposition.

Proposition 3.3: The steady state of the dynamical system (3.48)—(3.49) is globally asymptotically stable (totally unstable) if i� s

�(i� s

�).�

Proof: Calculating the time derivative of H along the trajectoriesof (3.48) and (3.49) yields:

H� �� h�(l)l1 � h�(�)�� h�(�) (

�� 1)�

�(i( · )� n� s( · )).

If i� s�holds, we get that the slope of i( · )� n� s( · ) is positive

�� Together with �� 1, of course.

� The dynamics are of Goodwinian type if i� s�holds.


(� (� i� s�)l). Furthermore i( · )� n� s( · )� 0 at � ��

�, i.e., this ex-

pression is negative to the left of��and positive to its right. The same holds

true for the function h�(�)�� i( · ) which, taken together with the previousresult, impliesH� � 0 for� ��

�. The assertion then follows from the usual

theorems on Liapunov functions, for which we refer the reader to Hirschand Smale (1974, pp.196ff.), and Brock andMalliaris (1989; pp.89ff.). In thesame way one can show H� � 0 if i� s

�.�

Up to now we have made use of linear relationships in the market for laboras well as for goods to investigate Rose’s (1967) broader view on real wagedynamics.We have obtained a result similar to his, namely, that the steadystate will be locally unstable if investment reacts more sensitively to realwage changes than total savings. In this case a drop in real wages will createextra goods demand pressure and thus extra inflation, which will induce afurther fall in real wages and thus destabilize the neutral closed orbitstructure of the Goodwin model. This locally explosive dynamical behav-ior is turned into global stability in Rose (1967) bymeans of an appropriatenonlinearity in the excess demand function of the labor market and bymaking use of neoclassical smooth factor substitution. In Flaschel andSethi (1996), it is shown how this strategy can be applied to the presentcontext. Here, however, we want to stick to fixed proportions in produc-tion and thus will have to introduce at least one further nonlinearity inorder to obtain Rose’s limit cycle result for a system of type (3.48)—(3.49).The nonlinearity that Rose uses in the labor market is a very natural one

if one takes into account the Classical nature of our general model and itsspecial cases. It is of the form�� displayed in figure 3.2.��According to this form the money wage will become very flexible farther

off the steady state (by way of a rising adjustment speed for larger devi-ations of the employment rate from its ‘‘natural’’ level 1). The proof ofproposition 3.3 immediately shows that this nonlinearity alone is insuffi-cient in successfully overcoming the total instability of the case where i� s

�holds. In fact, H� � 0 holds quite independently of the form of the Phillipscurve, as long as

�� 1 is true (while the case

�� 1 brings us back to the

closed orbit structure of the Goodwin model). The phase portrait of(3.48)—(3.49) for i� s

�is then easily shown to be of the type� displayed in

�� See also Akerlof and Stiglitz (1969, p.278) on this sort of Phillips curve.�� This law (see figure 3.2) replaces the linear function �

�(V� 1).

� The �� —isocline is given by:

l/b� 1�l

��

1� �

1� �

��(i( · )� n� s( · ))�

�l

��

1� �

1� �

��(s�� i)l�� const.�

� l/a,

and is thus a strictly increasing function of � for i� s�.


Figure 3.2 A nonlinear law of demand in the labor market

figure 3.3.�In the case

�� 1, the restricted phase diagram of figure 3.3 is again

filled with closed orbits as in the Goodwin model, while �� 1 yields

trajectories which point inwards with respect to these closed orbits fors�� i and outwards in the case s

�� i. Though the dynamicalmotion is thus

now restricted to a corridor around the steady-state value l�� l(V� 1), it

is not viable as we have just seen.In view of the shape of the �� 0 isocline,� and the mathematical

equation underlying it, it is natural to introduce a further nonlinearity, nowin the market for goods, in order to obtain global viability for the consider-ed dynamics, namely bymeans of investment behavior.Herewe assume thetype of nonlinearity displayed in figure 3.4a.Thus, though investment is more sensitive than savings with respect to

real wage changes around the steady state, the opposite is the case for

� Note here that the true upper bound on the variable� is given by ((y� �� t�)/y) x and notby x as in figure 3.3.

� �� 0 is horizontal if �

� 1 holds. Note the formal similarity to the Kaldor (1940) trade cycle model.


Figure 3.3 Implications of nonlinearity in the labor market

larger deviations of the real wage from its steady-state level ��. The phase

portrait in figure 3.3 is changed by these assumptions as shown in figure3.4b.We have added to this phase portrait one cycle of the closed orbit

structure of the Goodwin subcase ( �

� 1) of this two-dimensional dy-namical system, and will now show that the trajectories in the case

�� 1

point inwards with respect to each of these Goodwin cycles in the regionsto the left of � and to the right of �� . By contrast, they point outwardswithin these two values of �.

Proposition 3.4:We consider the Liapunov function of proposition3.3,

H(�, l)��

��

h�(�& )�&

d�& ��

��

h�(l� )

l�dl� ,

h�(l)� (1� �)��(l/l� 1), h�� i(�),

Note that the cycle is clockwise — and between the limits a, b — when the variables u,V areused in the place of �, l.


(a)

(b)

Figure 3.4 (a) A nonlinear investment-savings relationship; (b) a Rose limit cycle inthe fixed proportions case


but now augmented by the two nonlinearities just considered. We have inthe present case:

H� � 0 for all � �� or � �� ,

and

H� � 0 for � �� .

Proof: For � �� we have 0� i( · )� s( · )� n and �i( · )� 0,whilst for� �� we have s( · )� n� i( · )� 0 and�i( · )� 0 by assumption.The function H therefore fulfills

H� � (1� �)��(i( · )� n� s( · ))i( · )� 0 for all � �� and all

��

(and it is positive in between these bounds on�). Since we know thatH� � 0along the closed orbits of the Goodwin case

�� 1, we thus get that the

trajectories of the dynamical system (3.48)—(3.49), modified by the abovetwo nonlinearities, must point inwards along those segments of the Good-win cycle that lie outside of the interval (�,�� ).�

Any trajectory off the steady state consequentlymust cycle around it (sinceit has to stay inside of an appropriate Goodwin cycle when it leaves theabove depicted domain on its right hand side). It is, however, not yetexcluded that this occurs in an explosive fashion towards the boundaries ofthe domain depicted in figure 3.3.Assume now in addition that

�� 1 if�/x� (y� �� t�)/y, i.e., there is a

full cost-push effect of nominal wages with respect to the formation of theprice rate of inflation if real wages tend to eliminate profit income. The�� 0 isocline then tends to the horizontal line l/b as � tends to this limit.In this case we furthermore can state the following.

Proposition 3.5:The�-limit sets� of trajectories starting to the leftof (y� �� t�)x/y are all compact, nonempty and do not contain the steadystate (�

�, l�), i.e., by the Poincare—Bendixson theorem� they must be

closed orbits.

All trajectories which start to the left of (y� �� t�)x/y are thus attractedby some limit cycle within this set or are closed orbits themselves. This isillustrated by the simulation of the real cycle model displayed in figure 3.5,which is based on nonlinearities in the investment function and the Phillips

� The sets of all limit points of the considered trajectories.� See Hirsch/Smale (1974, p.248).


Table 3.1.

s�� 0.8, �� 0.1, y� 1, x� 2, l� 0.5, n� 0.05.

h�� 0.1, h

��, i� 1, �

�� 1.

�� 1, �

�� 1,

��

�� 0.5, �� 0, �� 0.

�� 0.05,

�� 0, t�� 0.35.

curve mechanism of the type

i( · )� atan(10�pi�(�� r� �))/(10�pi),X�� tan(1.25�pi�(V� 1))/(1.25�pi) for V� 1,X�� tan(2.5�pi�(V� 1))/(2.5�pi) for V� 1,

and on the set of parameters displayed in table 3.1.The steady state of this real cycle model is disturbed at time t� 1 by a

supply-side shock. Note here that the depicted limit cycle is based on thevariables u and V of the Goodwin growth cycle model and that the rangecovered by the variation of goods market excess demand allows for fourdifferent states. Note, furthermore that the loop showing up in the Phillipscurve in the lower right hand portion of figure 3.5 is clockwise and notcounterclockwise, as empirical observations have suggested.Note that in the above we have not provided a complete proof of

proposition 3.5, since we have only conjectured in the present situationthat all trajectories of this dynamical system can be continued withoutbound (and that they and their limit sets stay in the interior of theeconomically motivated rectangle depicted in figure 3.3). The applicationof the Poincare—Bendixson theorem is therefore not straightforward in thepresent situation. Such ambiguities can be avoided when the l/a-curve canbe shown to be (slightly) negatively sloped, as will be the case in the nextchapter and also in chapter 5 when we allow smooth factor substitution.The limit cycle approach of Rose’s (1967) employment cycle model thus

also applies to the present context and could be further investigated as inRose (1967). An important property of the above assumptions is that thedynamical behavior is thereby restricted to economically meaningfulvalues of �. Observe also that the problem encountered in section 3.3 withrespect to labor supply bottlenecks can now be completely avoided just bychoosing the parameter b in the Phillips curve of figure 3.2 such thatl/V

�� l/b holds true.

We have treated so far only the case K� � I (or l1 � � i( · )). The alterna-tive case K� �S (or l1 � n� s( · )) is similar and will give rise to the sameresults as l1 � � i( · ) (since n� s(�)� n� s

�(y� ��l� t�) is then of the

same qualitative form as the function �i(�)). Of course, an appropriately


Figure 3.5 The real cycle of the Keynes—Wicksell model

chosen nonlinear s( · ) function can also be used to investigate the dynami-cal behavior of (3.48)—(3.49) under such a modification.In sum, we may conclude that the Rose extension introduces local

instability into the Goodwin labor-market dynamics, but also provides themeans of establishing global stability, giving rise to a limit cycle resultinstead of the structurally unstable closed orbit structure of the Goodwinmodel. This is definitely an improvement over Goodwin’s growth cycleresult. The robustness of Rose’s employment cycle will be further inves-tigated in the following section.

3.5 Monetary growth cycles: the basic case

Removing one further assumption, namely, that concerning interest rateinflexibility in the real Goodwin/Rose growth cycle dynamics, we show inthis section that the now integrated real and monetary dynamics willsuppress the Rose employment (limit) cycle result with its instability of thesteady state, if the flexibility of nominal interest rates becomes sufficientlyhigh. This is mainly due to the Keynes effect which, as has often been


emphasized by static analysis, also fulfills its supposed stabilizing role in athree-dimensional dynamic growth context. The resulting asymptotic sta-bility of the steady state will, however, often rest nevertheless on cyclicaladjustment patterns.So far, we have only studied the cyclical properties of the real part of the

model by making it independent of money-market phenomena and expec-tations through appropriate assumptions on the interest rate elasticity ofmoney demand, on the adjustment of expectations, and on one furthersecondary assumption which, taken together, removed the influence ofmoney and bonds (expressed per unit of capital value), i.e., of the variablesm and b from real-wage dynamics and capital accumulation. In this section,we will now integrate the impact of the evolution ofm on the real dynamicsby allowing the interest rate r to fluctuate and by allowing to be positive theparameter

�, which describes the extent by which government expendi-

tures are money financed.The assumptions �� (�� ) and t�� (T� rB)/K� const. will,

however, still be made in order to allow inflationary expectations to holdstatic at the steady state value � �

�� n and, as always, for a treatment of

the model where bonds can remain implicit. Medium-run adjustments inexpectations will be considered in the next section.The model to be investigated in this section is thus given by the three-

dimensional dynamical system

�� [(1� �)��(V� 1)� (

�� 1)�

�(i( · )� n� s( · ))], (3.50)

V1 �K1 ( · )� n, (3.51)

m� �� p( · )�K1 ( · ), (3.52)

where

K1 ( · )� i( · )� n (or s( · )),i( · )� i(y� � ��l� r�

�� n),

s( · )� s�(y� � ��l� t�)� (g� t�),

g� t��m, t�� const.

r� r(m)� r�� (h

�y�m)/h

�, r�� 0 and

p( · )� �� n� [�

�(i( · )� n� s( · ))�

��(V� 1)].

With respect to this model we are able to prove the following propositionwhich asserts that flexibility of the nominal rate of interest of a sufficiently

Rose (1967) assumes for the followingmonetary extension of his model of the employmentcycle the relationship r� r(y) (with a variable ratio y due to the existence of neoclassicalfactor substitution), which allows — as in his paper — a reduction of the dynamics again todimension two in the two real variables � and l. In the context of the present dynamicalmodel this does not represent, however, a convincing simplification.


high degree will remove the Rose-type local instability from the real part ofthe model and thus also the possibility of it generating an employmentlimit cycle.

Proposition 3.6: The steady state of the dynamical system (3.50)—(3.52) is locally asymptotically stable if �r�(m

�)� 1/h

�is set sufficiently

large.

Proof: (for the case K1 � i( · )� n): For the Jacobian J of thedynamical system (3.50)—(3.52) at the steady state we obtain

J�� (

��1)�

�l(s

�� i)�

� (1�

�)��

� (

�� 1)�

�(� ir��

�)�

�

�ilV�

0 �ir�V�

ilV��

�(s�� i)lm

��

��m

�ir�V

��

�(� ir��

�)m

�� .

By means of the standard rules for the calculation of determinants, thedeterminant of J is easily shown to be equal to

�J �� (

�� 1)�

�(s�� i)l�

� (1�

�)��

� (

�� 1)�

�(� ir��

�)�

�

� ilV�

0 �ir�V�

0 � ��m

��1�

�1�

�

��m

�0

��

��m

��

1� �

1� �

��m

�� (

�� 1)�

�(s�� i)l�

� (

�� 1)�

�(� ir��

�)�

�

�ilV�

�ir�V� �

� � ��

� �� 0

This result also holds for �� 0 and it is independent of the size of r�. This

is the first of the four Routh—Hurwitz conditions (see Brock and Malliaris1989, pp.75ff., and the appendix to section 1.8), which are necessary andsufficient for the local asymptotic stability of the steady state.The next condition demands that the sum of the leading principal

minors: J��J

��J

of the above Jacobian must be positive. Due to the

‘‘0’’ in the middle of the Jacobian J, this positivity is obviously true for J�

and J. For J

��

J��

J�

J�

J� we obtain

J��

( �

� 1)��l(s

�� i)�

� (

�� 1)�

�(� ir� �

�)�

�ilV

�ir�V

� ��

( �

� 1)��l�

� (

�� 1)�

��

�ilV

�ir�V

� �


� ��

� � �� 0

This result also holds for �� 0 and it is independent of the size of r�.

The third condition is trace J� 0. We calculate

trace J� ( �� 1)�

�l(s

�� i)�

�� ir�V

��

�(� ir� �

�)m

�.

The condition trace J� 0 is obviously fulfilled when Rose’s model islocally asymptotically stable (i� s

�), and it can always be fulfilled in the

opposite case (i� s�) if r� is chosen sufficiently large.

The final Routh—Hurwitz condition is (� trace J)(J�� J

��J

)�

detJ� 0. To see that this condition can be fulfilled for derivatives r�(m�)

which are chosen sufficiently large in absolute value it suffices to note that(� trace J)(J

��J

��J

) is a quadratic function of r�, whereas det J

depends only linearly on it. The sign structure of trace J and J�, J

�, J

we

have discussed above then implies that b must become positive for suffi-ciently large values of � r� � .�

In the following proposition we establish that a limit cycle is born as r�(m�)

decreases in value.

Proposition 3.7: There exists exactly one value of r�(m�) (denoted

r�(m�)�) such that the steady state is unstable for r� in (r�(m

�)�, 0) and stable

in (��, r�(m�)�). At the value r�(m

�)� a Hopf bifurcation occurs, i.e., in

particular, the stability proven for large � r�(m�) � is lost in a cyclical fashion

as r�(m�) increases across this bifurcation value.

Proof:The proof of proposition 3.6 has shown that we have for thequantities a

�� trace J, a

��J

��J

��J

and a

� �det J the rela-

tionships

a��

�� r�(m

�) ��

�, (�

�� 0),

a��

�� r�(m

�) ��

�, (�

�� 0),

a� �

� r�(m

�) � , (�

� 0).

The polynomial b( � r�(m�) � )� a

�( � r�(m

�) � )a

�( � r�(m

�) � )� a

( � r�(m

�) � ) must

be quadratic and bear to the linear function a�( � r�(m

�) � ) the relationship

shown in figure 3.6.We know that there exists a unique � r�(m

�) � where a

�� trace J will be

zero. It follows that b must be negative at this value of � r�(m�) � , since

a�� det J is positive throughout.We thus get that a

�,a

�, a

, and bmust

all be positive to the right of � r�(m�)� � in the figure 3.6. This proves the first


Figure 3.6 The two Routh—Hurwitz coefficients a�, b

part of the proposition, since b cannot become positive again for lower� r�(m

�) � before a

�has turned negative.

The second part of this proposition can be proved as in the proof of aHopf bifurcation for the general Tobin model considered in Benhabib andMiyao (1981).�

This last proposition tells us that, at least in a certain neighborhood ofr�(m

�)�, the dynamical behavior of (3.50)—(3.52) must therefore be of a

cyclical nature.We know, furthermore, from the preceding section that it isof this same kind also for values of r�(m

�) sufficiently close to ‘‘0.’’ It can

therefore be expected that the model gives rise to monotonic adjustmentpaths to its steady state, if at all, only if r�(m

�) is sufficiently close to ��.

The proof of propositions 3.6 and 3.7 for the case K� �S is similar. TheHopf-bifurcation theorem can furthermore also be applied to the par-ameters �

�and �

�, and will give rise to similar propositions depending

upon the influence of the real wage on excess demand in the market forgoods. This issue is discussed further in the next chapter.In sum, we have so far found that moneywage flexibility and interest rate

flexibility work in favor of economic stability, while price flexibility gen-erally works against economic stability.


3.6 Expectations and the pure monetary cycle

Due to our formulation of inflationary expectations (3.24), we have thechoice between adaptive, regressive,� and myopic perfect foresight expec-tations (or a combination of these). As we shall see, regressive expectationspreserve the stability properties of the model of the preceding section, whileadaptively formed expectations can, when sufficiently fast, destabilize thedynamics through the working of the Mundell effect. Myopic perfectforesight expectations can be treated as the limit case of adaptive expecta-tions and thus face the same instability problems as fast adaptive expecta-tions. Furthermore, there are economic reasons why this situation ofmyopic perfect foresight should be excluded from our models in theirpresent formulation and analysis should be restricted to situations whereboth forward and backward looking behavior prevail.After having considered (in the following discussion) various special

cases of expectations formation, we shall then apply the forward andbackward looking expectations mechanism to an investigation of themedium run. In this medium run factor growth is ignored on the side ofproduction and real wage changes are suppressed by means of the twoassumptions �

�� 0,

�� 1, i.e., nominal wages are of an extremely slug-

gish type with respect to demand pressure on the labor market and theactual rate of inflation has a full impact effect on nominal wage formation.These assumptions result in a monetary dynamics subsector of the Cagantype, i.e. the isolated dynamic interaction between the two variablesm and�. Such a system of monetary dynamics has been often studied in theframework of pure money-market adjustments under adaptive expecta-tions as well as under perfect foresight.�Here, however, we shall considerthe product market and its adjustments instead and the influence of anadditional variable, the nominal rate of interest, which is determined bymoney-market equilibrium. This situation will give rise to a pure monetarylimit cycle in the above two variables if the nonlinear investment functionof section 3.4 is again assumed to apply.Because of the applicability of the assumption for the generation of real

limit cycles, (see figure 3.4) also to the generation of monetary cycles it isobvious that these two cycle models can be coupled with each other if theabove two assumptions on �

�and

�are relaxed. This coupling of the real

with monetary cycles will be briefly investigated in section 3.7 by means ofcomputer simulations.Our analysis in this section proceeds by analyzing various limiting cases

� See appendix 2 of chapter 4 for an extension and alternative explanation of thisforward-looking component of inflationary expectations.

� See Chiarella (1990, ch. 7) and Turnovsky (1995, ch. 3).


of the expectations mechanism and the limiting case of infinite speed ofprice adjustment.We first consider regressive expectations by setting �� 0, ��. In

the case of purely regressive expectations, the 3 3matrix J in the proof ofproposition 3.6 is augmented by a fourth column and a fourth row, thelatter being represented by

(0 0 0 ��)

since the new fourth dynamical law is here simply given by �� (�

� n��). We thus can state the following.

Proposition 3.8: The local stability properties of the four-dimen-sional dynamical system under regressive expectations are the same asthose of the dynamical system (3.50)—(3.52) considered in section 3.5.

Assuming purely regressive expectations thus does not add very much tothe analysis of section 3.5, the main difference being that inflationaryexpectations now slowly adjust to any new steady state value of

�� n,

while they immediately jump to it in the cases we investigated previously.We consider next adaptive expectations by setting �� 0,�� . In

the case of adaptive expectations the resulting four-dimensional dynamicalsystem becomes fully interdependent, since at least the evolution of � andm depends on � and that of � on the evolution of all three other dynamicvariables. The evolution of inflationary expectations � is now determinedby �� (p��), where p�� [�

�i( · )� n� s( · ))�

��(V� 1)].

This gives for the dependence of � on itself the expression �� /�� i�

� 0, since i( · ) (but not s( · )) depends positively on inflationary expectations�. This expression (�J

of the Jacobian of this extended dynamical

system) shows that the model of section 3.5 can always be made locallyunstable by choosing the parameter �� sufficiently high. As is known fromother models, we here recover the result that adaptive expectations create,at least locally, explosive behavior if they become sufficiently fast. On thebasis of the foregoing observations we state the following.

Proposition 3.9: The trace of the Jacobianmatrix J can be made aspositive as desired by choosing the adjustment parameter �� sufficientlylarge.

We conjecture that the loss of stability that comes about by increasing ��from ‘‘0’’ to ‘‘��’’ will occur again in a cyclical fashion bymeans of a Hopfbifurcation, as was the case in the previous section.�

� See Flaschel (1993, ch. 6) for investigations of a related situation.


Next we consider myopic perfect foresight by setting �� 0,�� .The fact that the trace of J approaches �� for �� in the just-considered case of adaptive expectations indicates that the limit case��, i.e. � � p, may be of a problematic nature. In this case, the twoPhillips-type adjustment mechanisms (3.22) and (3.23) of our generalframework reduce to

�� (V� 1), (3.53)

��

�((I�S)/K), (3.54)

and thus give rise to two different and seemingly contradictory real wagedynamics if

�� 0 and �

�� holds true, unless labor-market disequilib-

rium V� 1 and goods market disequilibrium are always proportional toeach other bymeans of the factor��

�/(�

� �). Under this side condition the

model is of the form (in the case K� � I)

�� (V� 1), (3.55)

V1 � i(�(�)� r(m)� p), (3.56)

m� �� p� i(�(�)� r(m)� p)� n, (3.57)

where p has to be calculated from

��(V� 1)��

�(i(�(�)� r(m)� p)� n� s

�(y� ��l� t�)

��m).

This gives for p the expression

p� [� �(��/��)(V� 1)� s

�(y� �� l� t�)

��m� n]/i� �(�)� r(m). (3.58)

In the special case �� 0� (and

�� 0), which implies I�S or

i( · )� n� s( · ), this determination of the rate of inflation p reduces to�

p� (s�(y� ��l� t�)� n)/i��(�)� r(m). (3.59)

We then get for the second of the above three laws of motion

V1 � s�(y� ��l� t�)� n, (3.60)

and thus again the simple growth cycle model (which we have investigatedin section 3.3) as far as the real dynamics (�,V ) is concerned. For the third

� Or �� , see the following.

� Such a situation is investigated in Sargent (1987, ch. 5) for the case of a constant value of gand

�� 0 by means of the saddlepath methodology introduced in Sargent and Wallace

(1973). See the following for further discussion of this methodology.


law of motion, which does not feed back into the real part of the modelunder the assumed circumstances, we furthermore obtain

m� m(�,V,m) with m�� 0, (3.61)

which gives rise to the saddlepoint instability situation to which theSargent and Wallace (1973) jump-variable methodology is then generallyapplied in the literature.Yet, the question remains, whether the adaptive expectations case

should not be reformulated first in such a way that it gives rise to a viabledynamics also in the case of a fast adjustment of adaptive expectations.Otherwise, there is the danger that the perfect foresight limit just formallyinherits economically implausible reaction patterns of the adaptive expec-tations case which are, in the case of myopic perfect foresight, then hiddenin the algebraic conditions to which the equation � � p gives rise. In thisregard, a plausible alternative to the conventional saddlepath procedurecan be obtained by nonlinear modifications of the adaptive case and theconsequent limit cycle and limit limit cycle results in the simple Caganframework of Sargent andWallace (1973) as expounded by Chiarella (1986,1990, ch. 7) and Flaschel and Sethi (1999).We finally consider forward and backward looking expectations by

choosing �� (0,�),�� (0,�). This case formally represents the summa-tion of the case of adaptive and regressive expectations and it thus inheritsthe stability and instability features of its two borderline cases we have justdiscussed. Note here that this combined situation can also be expressed as

�� (�� )[�p� (1� �)(�� n)� �], ��

��

. (3.62)

This form states that a certain weighted average of the currently observedrate of inflation and of the future steady-state rate is the measure accordingto which the expected medium-run rate of inflation is changed in anadaptive fashion.Note also that the actual rate p can be interpreted as myopically forward

as well as backward looking as long as the adjustment speed ��of prices p

stays finite, i.e. as long as prices are a differentiable function of time. Thismeans that the above formula can also be interpreted as being forwardlooking in both of its measures of the short and the long run. Again, it thenmeans that expectedmedium-run inflation is changed in the direction of anaverage of these two measures of inflation.Stressing the present mixed case of expectation formation as the truly

See Groth (1988) for further details on the discussion and analysis of such a combinedmechanism and appendix 2 of chapter 4 for the introduction of a more elaborate definitionof the forward looking component of this mechanism.


general one thusmeans that we insist on a proper combination of short-runand long-run information in the determination of the evolution of theexpected rate of inflation that is used in our expressions for the formationof planned investment, wages as well as prices. We recall that these aregiven by

i( · )� i(�(�)� (r� �)),��

�( · )�

�p� (1�

�)� ��

�( · )�

�(p� �),

p��( · )�

�w� (1�

�)��

�( · )�

�(w��).

Myopic perfect foresight may be considered to be included as a limitingcase in the last two equations, but should not be identifiedwith the rate � asin the above considered one-sided myopic perfect foresight case, since thiseliminates an important economic distinction in the present model (be-tween the rates p and �) and also introduces strange implications as wehave seen above (see equations (3.53) and (3.54)). Corresponding to themedium-run character of the rate �, one has to interpret the measureM ofthemoney supply in a broader sense in order to relate the determination ofthe nominal rate of interest also to the medium run.We do not consider in this book the extension just discussed in order to

ensure that the dynamical system brought about by the wage—price sectornot be of too high a dimension. Improvements in the formulation of thissector would therefore still be helpful in showing that the situation whereonly myopic perfect foresight prevails (and nothing else) should be con-sidered as too exceptional for a representation of the wage—price dynamicsof complete models of monetary growth.Keynes—Wicksell models have not really been considered in the litera-

ture on descriptive monetary macrodynamics, even on the textbook level.Their limit case �

��(I�S), which is usually based on a neoclassical

production function (see chapter 5, section 3), is, however, generally takento represent the Keynesian variant of the neoclassical synthesis and thusviewed as underlying the widely accepted Keynesian AS—AD formulationof monetary growth dynamics as discussed by Sargent (1987, ch. 5), forexample. We here show that the resulting model is nevertheless a purelysupply side model of monetary growth and thus demonstrate that the label‘‘Keynesian’’ for this type of growth dynamics is totally misleading.There, in fact, does not yet exist a proper formulation of ‘‘Keynesian’’monetary growth dynamics in all those model variants that start fromPatinkin’s (1965) neoclassical synthesis in their formulation of monetarygrowth. Such models are generally developed by simply adding nominalwage rigidity to the Patinkin formulation of the full employment case.

See chapter 5 for more details on this assertion.


As just stated, our general framework (3.1)—(3.24) of Keynes—Wickselltype has remained alive mostly through textbook presentations of thespecial case �

�� of AS—ADgrowth, i.e. by the case where goods-market

equilibrium prevails at all moments of time. This model is usually charac-terized as representing ‘‘Keynesian dynamics,’’ see Turnovsky (1977, ch. 8,1995, ch. 2) or Sargent (1987, ch. 5), for example. By assuming goods-market equilibrium throughout, the Wicksellian theory of inflation is onlypresent in the background of the model and, if at all, only consideredexplicitly as an ultra short-run adjustment mechanism, as in Sargent (1987,ch. 2).It is obvious from our above discussion of the case of myopic perfect

foresight that the model is then (for �� 0) of a purely Classical Goodwin

growth cycle type in the case of market clearing prices p(�� ), since we

then simply get as the dynamics for the real sector the two differentialequations,

�� (V� 1),

V1 � s�(y� ��l� t�), l� y/x,

whereas for themonetary part of the model we obtain by way of the IS—LMequilibrium conditions the single nonautonomous differential equation

m(t)� � r(m(t))� f (t),

where f (t) collects the dynamics of the predetermined real variables in-volved in the IS—LM equations.An infinite adjustment speed of the price level with respect to (potential)

goods-market disequilibrium combined with myopic perfect foresight thusgives rise to the same situation as we obtained above for the case where thegoods market was forced into equilibrium by assuming

�� 0 and myopic

perfect foresight. In both cases we have goods-market equilibrium on thebasis of a full utilization of the capital stock at eachmoment in time, so thathere nothing is left from the Keynes part of this model type. This issue isdiscussed further in chapter 5.This degeneracy of the model for an infinite adjustment speed of the

price level p is less obvious in the model with adaptively formed expecta-tions, which in the case

�� 0� is described by the differential equations

�� (V� 1)� (p��),

V1 � i(�(�)� r��),�� (p��),

where r and p have to be determined from the equations for IS—LM-

� This is generally assumed in such ‘‘Keynesian’’ IS—LM growth models.


equilibrium. The Classical nature of this particular IS—LM-equilibriumversion of the Keynes—Wicksell model also becomes obvious, however,when it is realized that such models always assume that the capital stock isfully utilized. In the present case this then gives rise to the equations (wecontinue to assume

�� 0)

i(�(�)� r��)� n� s�(y� ��l� t�), y, l� const.

m� h�y� h

�(r�� r), y� const.,

the first of which gives the rate of interest r as a function of the real wage �and expected inflation �(r� r(�,�)), while the second one then determineson this basis real balances per unit of capitalm (and thus implicitly the pricelevel p and its rate of change p).The foregoing analysis is, however, a very Friedmanian usage of the

IS—LM block of such a monetary growth model. It makes the abovedynamical system a three-dimensional one, since both r and p can beexpressed solely as functions of �, V, and �.The general conclusion here is that the IS—LM-equilibrium subcases of

our general Keynes—Wicksell model do not become strictly Keynesianmodels simply by assuming I�S in the place of I� S, but instead owetheir characteristic features still to the Classical nature of this Keynes—Wicksell approach to economic dynamics. This topic is reconsidered insection 3 of chapter 5.For the remainder of this section we assume, on the basis of the above

discussion, that the parameter values ��, ��, and �� are all positive and

finite. We thus exclude the one-sided cases we have considered above fromthe following discussion of the interaction of expectations first with theprice dynamics and then with the real cycle of the model. We here alsoassume

��

�� n for reasons of simplicity.

In order to derive the pure form of the monetary cycle in this case weshall make the following two sets of assumptions:

(i) ��

� 0, �� 1, so that the real wage is made a constant and in addition

is set equal to its steady state value,(ii) K� � n(L� � n), in which case the additional capacity effects of invest-

ment that are caused by profitability differentials (but not its trendcomponent) are suppressed on the supply side of the model (and onlythere). The labor intensity l�L/K thus is a constant in the followingand is set equal to its steady state value l in addition.

Both sets of assumptions can be justified in the usual way by stating thatthe intent of the present investigation is confined to some sort of puremedium-run analysis. They here simply serve to reduce the dimension ofthe above-considered dynamical system by two to a two-dimensional one


in the variables m and �. The resulting dynamical system reads�

m� �� n��

�(i( · )� n� s( · )), (3.63)

�� (i( · )� n� s( · ))��(�

� n��), (3.64)

where

i( · )� n� s( · )� i(�� r(m)� �)� n� s�(�� t�)� nm� g(m,�),

with g�

� 0, g�� 0. The isoclines m� � 0,�� 0 of the above two-dimensional dynamical

system are implicitly defined by

0�� n� ��

�g(m,�), (3.65)

0�� g(m,�)� ��(�

� n��). (3.66)

Equations (3.65) and (3.66) are globally well defined functions m of �. Thefunction defined by (3.65) has slope

m�(�)��

�g�� 1

��g�

.

On the other hand the function defined by (3.66) has slope

m�(�)��

�g�

�� g�

.

These two expressions immediately show that the slope of the first isoclineis always negative and smaller than the slope of the second isocline. Thelatter slope is positive far off the steady state (for positive values of theparameter ��), but may become negative in a certain neighborhood of thesteady state if a nonlinear shape for the investment function is assumed asin section 3.4 (see figure 3.4) and if the sizes of the various adjustmentspeeds are chosen appropriately. This follows immediately from the rela-tionship g�� i�( · ) and the fact that the slope of the investment functionbecomes zero far off the steady state by assumption.The phase portrait of the above dynamics of dimension two may there-

fore appear as in figure 3.7. Such a phase portrait can be easily tailored foran application of the Poincare—Bendixson theorem such that the non-

� Note that the rate of profit is constant in the present context. It is easy to show in the case of the above two-dimensional dynamical system that the

conditions for the Hopf-bifurcation theorem will apply with respect to the parameter ��.The following demonstration of the conditions that imply the validity of the Poincare—Bendixson theorem, however, gives rise to a situation that is much more general than thatof a Hopf limit cycle (or that of a Hopf closed orbit structure).


Figure 3.7 Phase diagram of the pure monetary cycle

negativity of the nominal rate of interest is assured.� To this end, one onlyhas to choose the parameter �� sufficiently large so that the isocline �� 0cuts the horizontal parts of the box depicted in figure 3.7 (the position ofthe other isocline is independent of this parameter).The derivation of limit cycle results is therefore in the present purely

monetary situationmuch easier than in the case of the real cycle consideredin section 3.4, but it obeys the same principles as were used there to obtainsuch a result. Figure 3.8 shows a simulation of this application of thePoincare—Bendixson theorem. Note that the excess demand contourshown in figure 3.8 (top right) is now strictly decreasing, since the savingscomponent in the excess demand function is constant here. The data forthis simulation are displayed in table 3.2.

� As long as m lies below the value m� at which r� r(m) is zero.


Table 3.2.

s�� 0.8, �� 0.1, y� 1, x� 2, l� 0.5, n� 0.05.

h�� 0.1, h

�� 0.2, i� 1, �

�� 1.

�� 0, �

�� 1,

�� 1,

�� 0.5.

�� 0.6, �� 0.15.��

�� 0.05, �

��

�� 0, t�� 0.35.

Figure 3.8 Simulation of the pure monetary limit cycle

3.7 The real and the monetary cycle in interaction

We have considered in section 3.5 the local Rose-type instability that iscaused by a negative dependence of goods-market disequilibrium on thereal wage (i� s

�) which, when coupled with a sufficient strength of speed of

adjustment of prices, gives rise to a positive dependence of the time rate ofchange of real wages on their level. Let us call this situation, in whichp�(�)� 0, a positive Rose effect for simplicity. In addition, we have inves-tigated in section 3.6 the local instability of the pure monetary mechanism


that is caused by the positive Mundell effect in the investment function(p�(�)� 0). These two destabilizing mechanisms, and the ways in which welimited their instability potential, will be integrated in this section byallowing for their full dynamic interaction in four dimensions.Before we turn to this topic, let us briefly explain why proposition 3.6

(where we had �� and ��) must also hold for all �� 0chosen sufficiently small and �� . This result follows from the follow-ing three observations; (i) the four-dimensional situation with �� 0 and�� 0 applied to this proposition exhibits three eigenvalues with nega-tive real parts and one further eigenvalue which is zero; (ii) the determi-nant of the Jacobian at the steady state of the dynamics (the product ofthe eigenvalues) is positive for all �� 0; and (iii) eigenvalues dependcontinuously on the parameters on the dynamics. The case �� 0, suffi-ciently small, is therefore characterized by at most two complex eigen-values with negative real parts and one negative eigenvalue, as in thesituation described in proposition 3.6, and one further negative eigenvaluewhich is close to zero.It is easy to show in addition that the stability just demonstratedmust be

lost if the parameter �� is made sufficiently large (since J� 0 is therebymade the dominant expression in the trace of the matrix J). Since thedeterminant of the Jacobian at the steady state is always positive, we inaddition know that this loss of stability will occur by way of a Hopfbifurcation, i.e., by way of the ‘‘death’’ of an unstable limit cycle or by wayof the ‘‘birth’’ of a stable limit cycle. From the local perspective we thereforeknow that the four-dimensional dynamics exhibits cyclical behavior atleast for a certain range of values of the parameter ��.In figures 3.5 and 3.8 we have furthermore considered the real cycle and

the monetary cycle (each in two dimensions) from a global perspective byadding a typical nonlinearity to the investment function of the model. Thisallowed us to apply the Poincare—Bendixson theorem to these two situ-ations and to conclude that there will be persistent fluctuations in the realand the monetary parts of the model whenever its steady state is locallyunstable, and that the two submodels are viable ones in a certain domain oftheir state variables.Since these two cycle mechanisms have been based on the same non-

linearity (in the investment function), we are interested in studying howthey interact on the basis of this viability generating nonlinearity. For themoment, however, this can only be answered by means of numericalinvestigations, an example of which is presented in what follows.�The following simulation displayed in figure 3.9makes use of a nonlinear

� This coupled growth cycle model can be usefully compared with Hicks’ (1974) analysis ofthe real and monetary factors in economic fluctuations.


Table 3.3.

s�� 0.8, �� 0.1, y� 1, x� 2, l� 0.5, n� 0.05.

h�� 0.1, h

�� 0.2, i� 1, �

�� 1.

�� 0.1, �

�� 2,

�� 0.95,

�� 0.5, �� 0.9, �� 0.4.

��

�� 0.05, �

��

�� 0, t�� 0.35.

Figure 3.9 Coupled real and monetary oscillators

investment function given by

i( · )� atan(10�pi�(�� r� �))/(10�pi),

which has the shape discussed in section 3.5. For the Phillips curve we takethe asymmetric shape given by

X�� tan(1.25�pi�(V� 1))/(1.25�pi) for V� 1,X�� tan(2.5�pi�(V� 1))/(2.5�pi) for V� 1.

The parameter values for the simulation are set out in table 3.3.


The steady state of this economy is disturbed at time t� 1 by a labor-supply shock. As can be seen from figure 3.9, the real cycle and themonetary one are here interacting with each other and are generatingsuperimposed fluctuations of a limit cycle type, with the monetary cyclebeing faster than the real one.Chiarella and Flaschel (1996a) show that more complex interactions

between the real and the monetary cycle of this chapter are possible. Theirsimulations indicate that the interaction of the two cycle generating mech-anisms may produce interesting phenomena, though not yet complexdynamics.

3.8 Outlook: less than full capacity growth

Some fundamental shortcomings of the neoclassical approach tomonetarygrowth of chapter 2 led us to develop the Keynes—Wicksell model of thischapter. Similarly, there is also a fundamental drawback of the present typeof monetary growthmodel that makes it absolutely necessary to revise andextend it further in order to get more descriptive accuracy for the modulesthat form the basic structure of a growing monetary economy. This basicdrawback relates to the feature (displayed by all monetary growth modelsof Keynes—Wicksell type) that firms always produce at full capacity, inde-pendently of the actual state of aggregate demand. Goods-market orIS-disequilibrium is thus always based on deviations of aggregate demandfrom potential output, and then used only to provide the measure for thedemand-pressure component of price dynamics or of the theory of infla-tion, and also to some extent in the equation describing capital accumula-tion.Yet, in the real world, labor and capital are often simultaneously un-

derutilized or overemployed, and in any case both generally not used on a‘‘normal’’ or ‘‘potential’’ level. Under- or overemployment is thus not onlya characteristic of labor supply but also of capital supply, and musttherefore be taken into account by any model of monetary growth thatattempts to be a generally applicable one.But how can a varying rate of utilization be introduced into the present

framework of a Keynes—Wicksell model of monetary growth? In the fol-lowing chapter this is done by assuming goods-market equilibrium in theusual way of an IS—LM approach to the determination of output andemployment, and thus of the rate of capacity utilization of both capital andlabor. Keynesian goods-market equilibrium is therefore now used, in theplace of IS-disequilibriumwithout themultiplier process, in order to deriveand explain the level of capacity utilization within the firm. Thismeasure of


disequilibrium is not accompanied in chapter 4 by disequilibrium outsidethe firm (on the market for goods).��Nothing unusual (in the sense of being novel) is therefore needed in order

to make the Keynes—Wicksell prototype model really a Keynesian modelof monetary growth with both labor and capital under- or overutilized. Inview of this obvious and natural modification of the Keynes—Wicksellapproach to monetary growth it is, however, very astonishing not to find inthe literature a well-established body of such Keynesian IS—LM growththeory. Orphanides and Solow (1990), for example, in their survey of theliterature on money, inflation, and growth, provide a brief survey onmodels of Keynes—Wicksell type, but do not mention any attempt toconstruct a growth model of the IS—LM variety.Going from Keynes—Wicksell to Keynesian monetary growth dynamics

therefore represents a crucial and compelling step in the further develop-ment of descriptive models of monetary growth. This step, which takes intoaccount that there is in general also disequilibrium within the firm, willnow be undertaken and considered in its implications. It will lead us inparticular to the result that the dynamics becomes much more differenti-ated through the new characteristic of the model that it is only the rate ofoutput, and not the price level and its dynamics, that is directly determinedthrough goods market (equilibrium) conditions. The dynamics of the pricelevel will now be derived indirectly and, in a second step, through thedisequilibrium level of capacity utilization as it is implied by goods-marketequilibrium (as the proper representation of the demand pressure compo-nent in the explanation of price inflation).Proceeding in this way the next chapter will therefore fill an important

and crucial gap in the literature on monetary growth, though it will not yetlead us, to a (final) basic working model of Keynesian monetary growthdynamics. This goal will be attained in chapter 6. Despite this intermediatecharacter of chapter 4 it will, however, clearly demonstrate that there issomething essential missing in the literature on Keynesian monetarygrowth. This missing element is not so much one with respect to specificand novel module formulations, but basically one with respect to anintegrated treatment and detailed analysis of module interactions of aproperly defined model of IS—LM growth.

�� The effects of such outside disequilibrium for the dynamics of themodelwill be added to themodel and be studied in chapter 6.


4 Keynesian monetary growth: themissing prototype

In this chapter, we shall further improve the general disequilibrium versionof the Tobin type models of chapter 2. We recall that this version wasalready considerably extended and improved in chapter 3 in the directionof more descriptive relevance and greater consistency as far as the treat-ment of basic disequilibrium situations in the real part of the economy wasconcerned. Of course, much still remains to be done in the pursuit of suchaims, and this will still be the case by the end of this book.� Nevertheless,the model prototype of this chapter represents for the first time a versionthat can be the basis of all future developments of the structure of suchmonetary growth models. It not only integrates problems of effectivedemand on the market for goods in a basically coherent way, but alsoportrays the fundamental consequences of such an integration in an integ-rated model of monetary growth.The prototype model we arrive at in this chapter can also be considered

to provide the minimal extension of monetary growth models of a moreorthodox type (as we have considered them in the preceding chapters) to aproper and basically complete Keynesian model of such an economy. Yet,due to this heritage, the model still exhibits many features which mayappear as questionable from a post-Keynesian and other perspectives. It is,however, our intention in this book to lay the foundations of a descriptiveKeynesian theory of monetary growth through a systematic increase in thecomplexity of such a model type by starting from conventional approachesto Keynesianism, to some extent even from the standard macroeconomictextbook level. Therefore, model substructures which may seem indispens-able from a more developed Keynesian perspective, for example ap-proaches which give asset markets a much more dominant role than wehave done so far, cannot be included until the scene is properly set for suchfurther developments. One of the main aims of this book is to achieve the

� See the survey of open problems in section 7.7.

173

proper setting of the scene. Hence a considerable number of typicallyKeynesian questions must here be left for future research.We are also convinced that a systematic treatment of Keynesian models

of monetary growth which starts from orthodox foundations is of value inits own right and will also help in the understanding of models which atfirst sight seem to be quite different in their structure. From this perspectiveit may be claimed that the present chapter is a core chapter for theintentions of this book.Section 4.1 will present the general, but still basic, Keynesian prototype

model and explain its structure.�Wealso derive in this section the intensiveform of this model and determine its unique interior steady-state solution.The simplified version of the model, where we neglect the GBR, will closethis section. In section 4.2 we provide some basic considerations of thetemporary equilibrium part of the model in general as well as of theimportant special case in which there is a given rate of interest. This specialcase is employed in section 4.3 from the perspective of (and as part of ) thereal growth/real wage dynamics of the Goodwin model, now with Key-nesian demand problems.We shall find that the Goodwin type dynamics ofthe preceding chapter (see section 3.3) are considerably changed by theintroduction of such demand problems. Section 4.4 then integrates theRose type influence of the market for goods on the formation of the realwage and considers from a local as well as from a global point of view howmuch of the stabilizing potential of flexible money wages (considered insection 3.4 for the Keynes—Wicksell prototype) will remain in the Key-nesian analog to this earlier section.The next section 4.5 then integrates monetary phenomena into these real

growth cycle models by way of money market dependent interest rateflexibility and its impact on investment behavior. As in the correspondingsection of the preceding chapter we demonstrate here a variety of local(in)stability results including the loss of asymptotic stability at certainparameter values of the model via the occurrence of Hopf bifurcations.Section 4.6 briefly expands the dynamic dimension of the model to four byallowing for adjustments in the expectations of medium-run inflation, andit considers the destabilizing potential of the Mundell effect in particular.Yet, such a higher dimensional analysis has to remain very preliminaryhere, though its integration of various cycle generating mechanisms ap-pears to be very interesting. Finally, section 4.7 briefly considers a limitingcase which is very important with respect to the existing literature onKeynesian dynamics, namely the case of Keynesian AS—AD growth whereit acquires the features of a supply side model and becomes closely relatedto results found for the Keynes—Wicksell prototype.

� See also Chiarella and Flaschel (1995) for another detailed discussion of this model type.


4.1 A general Keynesian model of monetary growth

4.1.1 Description of the model

We are now in the position to provide a prototype model of Keynesiantype on the same level of generality as the general Keynes—Wicksell modelof the preceding chapter. This model shows the minimum modificationsthat have to be made to this earlier model in order to transform it to onethat can really be considered Keynesian. That is, a model where output isdetermined by effective demand (here IS-equilibrium), which allows, forunderutilized capital as the new representation of goods-market disequi-librium, and which allows for underutilized labor. The utilization rate ofthe capital stock is here considered as the variable which adjusts immedi-ately to clear the market for goods ‘‘on the surface,’’ while the price levelwill respond to changes in capacity utilization only with a time delay(given by the reciprocal value of the adjustment speed �

�of prices p). Our

modeling procedure thus assumes that supplies (output) respond fasterthan prices (the price level), in the strict sense that the former variable isassumed as a statically endogenous one, i.e., capable of performing jumpsto restore temporary equilibrium, while the latter is assumed as dynami-cally endogenous, i.e., changes continuously in time according to its law ofmotion.Textbook and other presentations of the Keynesian model often also

treat the price level as a statically endogenous variable (determined by themarginal wage cost rule). We shall demonstrate in chapter 5 that theKeynesian model becomes a supply determined model when prices as wellas quantities adjust with infinite speed. Such a modification of the modelshould therefore be considered as one which leads us back to a(neo)Classical scenario and should consequently be excluded from the setof proper (i.e., demand determined) Keynesian models. One may neverthe-less insist that prices and quantities should be treated in a more symmetri-cal way, since prices and quantities may not differ by so much in theiradjustment behavior from wages as the very strict hierarchy assumedabove entails. This may indeed be a relevant proposal and it will be takenup in chapter 7, where we will assume that the adjustments of both outputand prices occur with a finite speed, just as wages, but may be very differentin size, nevertheless.Most of the following equations (and symbols) are the same as in the

Note here however that ‘‘output’’ Y is in fact ‘‘the rate of output’’ or ‘‘output speed’’ in acontinuous time model, and therefore has quite a different dimension as compared to theprice level p. Such a speed variable may change instantaneously (jump) without leading tojumps in the level of output (and the level of prices). Nevertheless, a Keynesianmodel shouldalso be capable of allowing for finite adjustments in output speed (see chapter 7).

175The missing prototype

Keynes—Wicksell model of the preceding chapter and will not be explainedhere once more. As already stated, the outputY of firms is now determinedby effective demand as in IS—LM models of textbook (or more elaborate)type. This means that we now have to distinguish between this output Yand potential output Y� as in equation (4.7). The resulting rate of capacityutilization, denoted byU, will exercise influence on investment behavior aswell as on price formation. These are the essential changes in going fromthe general Keynes—Wicksell model to the Keynesian one of this chapter.The extent of these changes may seem small, in particular when comparedwith the ones necessary to go from the Tobin case to the Keynes—Wicksellcase. But they definitely improve the consistency of the Keynes—Wicksellapproach significantly.One central question that arises is howmuchwill infact be changed in the quantitative responses of the system over timethrough the adoption of the indirect mechanism with respect to the conse-quences of goods-market disequilibrium (via the capacity utilization rateU) rather than the direct price mechanism of the Keynes—Wicksell model,which was based on direct goods-market disequilibrium at full capacityutilization. In particular we are interested in the effects of such changes oninflation as well as on investment.Note that we still have fixed proportions in production (y� and x are

given parameters), though output per unit of capital y is now a variable ofthe model. The case of smooth factor substitution will be treated in chapter5 for all three prototype models, where we will demonstrate in particularthat such an addition does not modify our prototype models, and inparticular the Keynesian one, in any essential way.As in the case of the labor market and the associated widely accepted

concept of a natural (or nonaccelerating rate of inflation) rate of employ-ment V� � 1, we now also assume as given a natural rate of capacityutilization U� � 1, which describes the benchmark between expansionaryand contractionary effects on prices and investment. We assume, further-more, that the distance of the maximum rate of capacity utilizationU

�� 1 from the natural rate of capacity utilization U� as well as the

distance of the actual rate of employment V from the maximum rate ofemployment (of the employed) V

�� 1 will both always stay positive so

that the maximum rates can be neglected as limits in all following investi-gations. Of course, supply bottlenecks, as they are generated by absolutelimits on the employment of the labor force L and by the capital supplyK,must be added eventually for a really complete analysis of the model byconsidering modifications of the assumed economic behavior when theeconomy comes close to or is at V

��,U

��. Neo-Keynesian regime switch-

ing approaches identifyU� with U��

and V� with V��

� 1, and thus claimthat the steady state path of a capitalistic economy is always right at the


border to such absolute supply bottlenecks. In our view, such assump-tions, however, do not represent a good description of the working of acapitalist economy.�

The equations of the model are:�


��w/p, u��/x,� � (Y� �K� �L)/K, (4.1)


%� 1. (4.2)



�pY� h

�pK(1� �)(r� � r), (4.3)

C��L� (1� s�)[�K� rB/p�T ], s

�� 0, (4.4)

S��L�Y�


� s�[�K� rB/p�T ]� s

�Y��

� (M� �B� � pE� )/p, (4.5)

L1 � n� const. (4.6)


Y�� y�K,y�� const.,U�Y/Y�� y/y�, (y�Y/K), (4.7)

L�Y/x,x� const.,V�L/L�Y/(xL), (4.8)

I� i�(�� (r� �))K� i

�(U�U� )K� K, (� n), (4.9)

pE� /p� I, (4.10)

K1 � I/K. (4.11)


T� �(�K� rB/p), [or t�� (T� rB/p)/K� const.], (4.12)

G�T� rB/p��M/p, (4.13)

S��T� rB/p�G [�� (M� �B� )/p, see below], (4.14)

M1 ��, (4.15)

B� � pG� rB� pT�M� [� (��

�)M]. (4.16)

We here instead use nonrepressed inflation and buffers provided by the behavior of firms inorder to avoid switches away from the Keynesian regime to so-called Classical regimes orregimes of repressed inflation, at least for a normal functioning of the economy around itssteady-state path; see Chiarella et al. (1999, ch. 3) for a further discussion of this topic.

� See chapter 7 for a further buffer that is created by capitalist firms in order to circumvent thecapital stock constraint.





�pK(1� �)(r� � r) [B�B,E�E], (4.17)

pE� (1� �)��K/((1� �)r��), (4.18)

M� �M� ,B� �B� [E� �E� ]. (4.19)

6 Equilibrium condition (goods market):

S� pE� � S

��S

��Y� �K�C�G� I� p

E� . (4.20)

7 Wage–price Sector (adjustment equations):

w� ��(V�V� )�

�p� (1�

�)�, (4.21)

p��(U�U� )�

�w� (1�

�)�, (4.22)

�� (p��)��(�� n� �). (4.23)

There is one central weakness and problematic feature in the formulationof the general Keynes—Wicksell prototype model of the preceding chapterwhich indeed disqualifies it as a candidate for a truly Keynesian dynamicmodel as (at least) its limiting case �

�� (augmented by a neoclassical

production function) is generally viewed and classified (see for exampleSargent 1987 andMcCallum 1989). This weakness stems from the fact thatthe firms of this model always operate at full capacityY�Y� (which is alsotrue for the case of smooth factor substitution we shall consider in chapter5), as if Say’s Law would hold true. Of course, there may be deficient orexcess demand in this model on the market for goods, but its effect solely ison the rate of inflation of this economy (if K1 �S/K holds), while there arein addition unplanned inventory fluctuationswith no further consequences(!) in the case whereK1 � I/K holds (see section 3.1 in chapter 3 for details).It is furthermore simply assumed in this Keynes—Wicksell prototype thatthese inventory fluctuations stay within such bounds that their influenceon the evolution of quantities can be neglected in this model.In the limit case �

�� of the Keynes—Wicksell model, this un-

Keynesian feature of the model becomes even more pronounced. It thengives rise to a model with full capacity utilization (but unemployed labor,due to nominal wage rigidities), where the Keynesian IS—LM equilibriumpart is solved through price and nominal interest rate adjustments andthereby adjusted to the predetermined supply of commodities in eachperiod. The model is then definitely following proposals made by Fried-man in the early seventies with respect to a monetarist reformulation ofIS—LM analysis, and may therefore be characterized as being of Wicksell—Friedman rather than of Keynes—Wicksell type.


Be that as it may, the inevitable conclusion is that this model must bereformulated in order to arrive at a Keynesian demand oriented modelwith its implications for both the utilization of capital as well as of labor. Afurther hint on the need for such a reformulation is given by the empiricalfact that severe disequilibrium on the side of capitalistic firms does not somuch show up in demand and (full capacity) supply imbalances for theirproducts as in the Keynes—Wicksell model, but rather in a severely un-derutilized capital stock, since production is relatively easily adjusted todeficient aggregate demand. The disequilibrium measure used in the Key-nes—Wicksell model of the preceding chapter (in the price adjustmentequation (3.23)) is therefore only appropriate when firms always operate atfull capacity (as was the case in that model type), but it is very implausiblein a model that allows for underutilized capital as a Keynesian modelshould do. The Keynesian model of this chapter therefore starts again from IS-

equilibrium, which makes the measure of goods-market disequilibrium ofthe preceding chapter completely irrelevant for the discussion of the deter-minants of the rate of inflation. However IS-disequilibrium will eventuallybe integrated into our ‘‘proper’’ (though still very simple) prototype ofKeynesian dynamics in chapter 6, where we shall further demonstrate thatthe Keynes—Wicksell treatment of it is indeed very misleading as far as the‘‘Keynes’’ label in denoting this model type is concerned.Starting from the general Keynes—Wicksell model type of section 3.1, the

derivation of a propermodel of Keynesianmonetary growth dynamics is infact not very difficult and demanding, so one may wonder why the proto-type model developed here is not yet a standard model of the macro-economic literature of Keynesian dynamics. An explanation for this fact isprovided in chapter 5, where it is in particular shown that the widespread(and unreflected) use of a neoclassical production function in combinationwith the Classical postulate on the equality between the actual marginalproduct of labor and the real wage in the conventional macroeconomicliteraturemay be responsible for the bastardmodel of Keynesian dynamicsso widely used in the literature. The exception to this characterization ofKeynesian dynamics is given by so-called fixed-price approaches where,

One essential difference of these dynamics to the dynamics of the general Keynes—Wicksellmodel is that the p dynamics are nowmore roundabout, since the IS part of the model nowno longer determines the rate of inflation directly, but instead determines the equilibriumoutput ratio y which then via the rate of capacity utilization determines the rate of inflation.Note here that low degrees of capacity utilization are muchmore plausible than large I�Sdiscrepancies as a general description of goods-market disequilibrium. Note also thatgoods-market disequilibria now have, and indeed must have, an impact on investmentbehavior. This is the other essential modification when going from the Keynes—Wicksell tothe Keynesian model.


however, the dominance of the so-calledKeynesian regime, as described byour prototype model, is not clearly established and where there is nofar-reaching and generally accepted prototype of monetary growth avail-able.The range of modifications leading from the Keynes—Wicksell prototype

to the above Keynesian one is in fact small, in particular when comparedwith the modifications that were necessary to go from the general Tobinmodel in section 2.5 to its Keynes—Wicksell extension in section 3.1.Equation (4.7) describes the basic new fact of the model in definitionalterms, namely, that this model now allows for varying degrees of utilizationU of the productive capacity of firms. This extension immediately gives riseto an extended formulation of investment behavior (again in a linear formfor the time being), which is now not only dependent on rate of returndifferentials, but also on the actual degree of capacity utilization. Theextent to which productive capacity is used depends here, as in standardtextbook presentations of Keynesian type, on the state of effective demandas it is determined by goods and asset market equilibrium (4.17)—(4.20).The final important modification of the model is given by equation

(4.22), which formulates a second type of Phillips curve, in addition to theproper Phillips curve (equation (4.21). This second Phillips curve relatesprice inflation with deviations of actual capacity utilization U from thedesired one, U� .� This desired rate of capacity utilization is exogenouslygiven and it plays the same role as the NAIRU rate of employmentV� in themany models of inflation theory that are based on such a concept (or evenon the so-called natural rate of (un)employment). The above symmetricformulation of wage and price Phillips curves has been fairly neglected inthe macroeconomic literature on inflation, due, on the one hand, to theidentity that is normally assumed between the rate of wage and of priceinflation (based on the original Samuelson and Solow 1960 assumption of asimple static markup theory of the price level) and, on the other hand, tothe assumed validity of Okun’s Law, according to which the utilization oflabor and capital are assumed to be positively related. This, however, neednot be the case in the current model, which demands that we have todistinguish between the above two Phillips curves (and their NAIRUs)from now on.� It would be interesting to have empirical observations onthis second type of curve in comparison to themany observations that exist

� Note that the present approachmeasures the actual rate of capacity utilization indirectly bymeans of the expression y�Y/K, implying that this second type of Phillips curve can easilybe estimated.

� Fair (1997a,b) has recently investigated such combinations of two Phillips curves, in asimplified aswell as in an extended form (that integrates technical progress in thewaywewilldiscuss it at the end of this chapter) from the empirical point of view for the USA andtwenty-seven other countries. These works thus provide empirical support for thewage—price dynamics as it is formulated in this chapter.


for its money-wage counterpart. We here stress once again that the abovetype of markup pricing behavior is also (at least partly) present in thisgoods-market reaction curve through the term

�w.

This concludes the description of the innovations of the present Key-nesian prototypemodel in comparison to itsKeynes—Wicksell predecessor.Note here that we assume for the time being that firms have perfectknowledge of the current aggregate demand schedule�� Y(Y, . . .), whichdepends on their output and employment decision and on which they basetheir market-clearing production decision Y�Y(Y, , . .), as well as theirdelayed price adjustment p� . . . as described by the price Phillips curve(4.22) with its demand-pressure and cost-push components (see Fair1997a,b for empirical tests of this Phillips curve for twenty eight countries).Based on such an empirically motivated price Phillips curve we are thusassuming that firms do their best with respect to the output decision theyare facing. We will allow for a simple error-correction model of salesexpectations in chapter 6 (section 3), where we describe the updating ofsuch point expectations in a way based onMetzler (1941), Franke and Lux(1993), and Franke (1996), and where we shall see that this makes theintegrated dynamics more complex, but also more robust.�� The goods-market equilibrium situation of this chapter can then be reestablished as alimit of this Metzlerian inventory adjustment mechanism, a limit case withsome peculiar properties, as we shall see in the course of this chapter. Notetherefore also that an explicit treatment of inventories is not necessary inthe present chapter since the level of production is always equal to the levelof aggregate demand. Note, finally, that we make use of the Keynesianregime of the regimes considered by non-Walrasian theory (see Malinvaud1980, for example, as far as the determination of the short-run level ofoutput is concerned). This can be and is justified by the buffers andadjustment processes that surround the steady state of our models ofchapters 4 to 7. The relevance of regime switches (towards the regimes ofclassical unemployment or repressed inflation, see again Malinvaud 1980),is investigated in Chiarella et al. (1999, ch. 3) for a general model of the typeof the working model of chapter 6 of this book and found to be ofsecondary importance compared to the Keynesian regime of demandconstrained output of firms.

4.1.2 Intensive form of the model and steady-state behavior

As far as the mathematical analysis of the model is concerned, we againconcentrate on the version which assumes that taxes net of interest pay-ments per value unit of capital are a given magnitude. The tax rate � can�� Given by Y/K��(L/K)� (1� s

�)(� � t�)� i

�(�� (r��))� i

�(Y/Y��U� )� n�

�� t�� (M/pK). �� See also Hahn and Solow (1995, p.7) in this respect.


then again be suppressed, in particular in the investment equation and inthe perfect-substitute assumption.Before we start the analysis of this special case t�� (T� rB/p)/

K� const., let us briefly present the general case of an endogenous deter-mination of this ratio and thus the existence of a feedback mechanism (stilla simple one!) of government debt accumulation B� on the rest of thedynamics. From calculations similar to those in the preceding chapter, weobtain from (4.1)—(4.21) the following autonomous five-dimensional dy-namical system in the variables ��w/p, l�L/K, m�M/(pK), �, andb�B/(pK):

�� [(1� �)��X�� (

�� 1)�

�X�], (4.24)

l1 � n� s( · )� � i�( · )� i

�( · ), (4.25)

m� �� n� [�

�X��

��X�]� l1 , (4.26)

�� [��X�� X�]��(�

� n��), (4.27)

b� � (��

�)m� (�� n)b� ( (�

�X��

��X�)� l1 )b, (4.28)

where we employ again the abbreviations (r� � r�again):

�� y� �� l, l�L/K� y/x (y not const.!),X��V�V� � l/l�V� � y/(xl)�V� ,X��U�U� � y/y��U� ,r� r

�� (h

�y�m)/(h

�(1� �))[m(y, r)� h

�y� h

�(1� �)(r

�� r),

see (4.3)],t�T/K� �(� � rb), t�� t� rb,g� t��

�m,

s( · )�K1 � s�(�� t�)� (g� t�)� I/K,

i�( · )� i

�(�� r��),

� (1� � �)��.

As new relationships, we now have in addition i�( · )� i

�X�, and the

following IS—LM determination of actual output per capital y� Y/K(� y��Y�/K):

s( · )� s�(1� �)(� � rb)�

�m�

i( · )� i�(y(1� u)� � � r� �)� i

�(U�U� )� n. (4.29)

We note that this equation can be solved explicitly for output per unit ofcapital y (see the next sections for some details on this determination of y).The steady state of this model is the same as that of the Keynes—Wicksellmodel described in section 2 of the preceding chapter, up to the obviousinfluence of the steady-state rate of capacity utilizationU� which is now lessthan 1.


Let us now again assume tn � t � rb � const. and remove the parameter� from the equations of the model (since taxes are now lump sum).��Furthermore, we set U� �V� � 1 for notational simplicity.� This givesagain a four-dimensional dynamical system in �, l, m, � with an appendedequation for the dynamics of b, since the influence of rb on s( · ) and g (andthus on y) is thereby suppressed. These dynamics will be investigated in theremainder of this chapter with respect to local as well as global stabilityproperties for a variety of subcases, following a sequence of generalizationssimilar to those of the treatment of theKeynes—Wicksellmodel in chapter 3.We assume in this analysis that the implied evolution for b will remain a

bounded one. This can be expected to hold at least for the case of anasymptotically stable steady state of the four-dimensional subsystem(4.24)—(4.27) for �, l, m, and �, since X�, X�, and l1 all tend to zero in thiscase. The dynamical behavior of b will then be dominated by b� �(

��

�)m� (�� n)b, with the steady-state value of b being given by

b�� (

�/

�� 1)m

�. Note here also that the coefficient�(� � n) of b on the

right hand side of this differential equation characterizes the entry J��of

the Jacobian of the full dynamics (4.24)—(4.28), and thus gives the fiftheigenvalue of this Jacobian in the uncoupled situation. The local stabilityproperties of the four-dimensional subsystem (4.30)—(4.33) below are there-fore augmented by the additional b equation and its eigenvalue in astraightforward way.There is again a unique steady-state configuration for the dynamical

system (4.24)—(4.28) with ��, l�,m� 0. In the present case with t�� const.

(r� � r�,U� �V� � 1) this steady state is given by

y�� y�, l

�� y

�/x, l

�� l

�,

��

�� n,

m�� h

�y�,

�� t��

n��m

�s�

,

�� (y

��

�)/l

�,

r��

��

�� n.

As in the preceding chapter, we assume that the parameters of the modelare chosen such that the last three steady-state values are all positive. Allfollowing analysis will be confined to stability investigations of or aroundthis steady state of the given model.

�� This is seen by noting from the above that � � rb� y� � ��y/x� rb and that�(�� rb)� t� t�� rb, hence (1� �)(� � rb)� y(1� u)� � � t�, where u��/x, theshare of wages in gross national product.

� This may be achieved through an appropriate redefinition of the sizes of the parametersx,�

�and y�, �

�.


Summarizing, the system (with t�� const.) that we shall investigate inthis chapter reads:

�� [(1� �)��(y/(xl)� 1)� (

�� 1)�

�(y/y�� 1)], (4.30)

l1 � n� s�(y(1��/x)� �� t�)��

�(y/(xl)� 1)�

�m, (4.31)

m� �� n� [�

�(y/y�� 1)�

��(y/(xl)� 1)]� l1 , (4.32)

�� [��(y/y�� 1)� ��(y/(xl)� 1)]

��(�� n��), (4.33)

where y is given by the solution of� (u��/x,x� const.)

s( · )� s�(y(1� u)� � � t�)�

�m�

i( · )� i�(y(1� u)� � � (r

�� (h

�y�m)/h

�)� �)

� i�(y/y�� 1)� n. (4.34)

In section 4.2 we shall briefly investigate the IS—LMsubsector of themodel,in particular in the case where the nominal rate of interest and the expectedrate of inflation are given magnitudes. There exist then three typicalsituations for the dependence of effective demand y on income distribution.These three cases give rise to three different phase diagrams when weconsider the simple Goodwinian interaction between real wage formationand capital accumulation in section 4.3, phase plots which no longer needbe very close to the original structurally unstable center type dynamics ofthe Goodwin model. The Rose extension of the real wage dynamics isconsidered next, in section 4.4, particularly in their potential to generatelimit cycles from the Goodwin overshootingmechanism still present in twoof the three cases. Section 4.5 extends to three the dimension of thedynamical system by again including the money market and interest ratephenomena.We shall see there that theKeynes effect will play a prominentrole in the return to asymptotic stability that turns out to be observed inthis case. This increase in the potential for stability is, however, againreduced or overthrown when adaptively formed expectations are added tothe model (in section 4.6) due to the destabilizing potential of the Mundelleffect.

4.2 Comparative statics: the IS–LM subsector

As already noted, the temporary equilibrium part of the model (in the caset�� const.) determines actual output per unit of capital y, and the rate ofinterest r, at each moment in time by means of equation (4.34) andr� r

�� (h

�y�m)/h

�. The implied equation for the determination of IS—

LM output thus reads:

� See equation (4.29) and footnote 12.


s�(y(1� u)� � � t�)� n�

�m� i

�(y(1� u)� �� (r

��

(h�y�m)/h

�)��)� i

�(y/y�� 1). (4.35)

Proposition 4.1: Equation (4.35) defines a function y(u,m, �) whichis locally well defined whenever the denominator in the following expres-sions for its partial derivatives is nonzero:

y"�

�y�u

�(s�� i

�)y

(s�� i

�)(1� u)� i

�h�/h

�� i

�/y�,

y��

�y�m

�� i

�/h

�(s�� i

�)(1� u)� i

�h�/h

�� i

�/y�,

y��y��

�i�

(s�� i

�)(1� u)� i

�h�/h

�� i

�/y�.

Proof: A routine exercise in comparative statics. Note that theabove equation (4.35) can also be solved explicitly with respect to thevariable y.�

The common denominator of the expressions proposition 4.1 shows thethree different forces that act on output per unit of capital y via investmentand savings behavior, i.e., the real-wage effect (s

�� i

�)(1� u), the capacity

effect�i�/y�� 0, and the interest-rate effect�i

�h�/h

�� 0. The sign of the

real-wage effect depends on the relative sensitivity of savings and invest-ment with regard to changes in the real wage. The sign of the denominatorcan be positive or negative, depending on the choice of s

�, i�, i�and h

�, h

�in

particular.Two cases will be of particular importance in the following, namely

h�� and h

�� 0. In the second case, the interest-rate effect dominates

the denominator and makes it unambiguously positive, if i�� 0 holds. We

then get:

Corollary 4.1: Assume that y� 0, u � (0, 1) is guaranteed and thath�is chosen sufficiently small. Then: y

"� 0 iff s

�� i

�, y�� 0, y� � 0.

The effect y�� 0 is called the Keynes effect in the literature, while y�� 0 is

the so-called Mundell effect. Increasing money supply and inflationaryexpectations thus both increase economic activity in this case.The first case h

�� is the (hypothetical) case of the liquidity trap we

have already made use of in the exploration of the most basic type ofdynamical behavior of the Keynes—Wicksell model. This case will beinvestigated again below as the starting case of our analysis of the dynami-cal system (4.30)—(4.33).


Note that the Keynes effect and the Mundell effect are reversed in sign ifthe denominator expression becomes negative, i.e., if s

�� i

�and h

�/h

�sufficiently small. Note that y

"will then generally be positive.

Note finally, that �"� y

"(1� u)� y can be negative (for 1� u� 0),

even if y"� 0 holds.�� Most of the ensuing investigations will rely on the

prevalence of the conditions �"� 0, y

�� 0 and y�� 0, i.e., on a normal

functioning of the profitability effect of real wage increases, the Keyneseffect and the Mundell effect (but with y

"� 0).

Let us now consider the case r � r0(h2��),��0� n(��2��),

which we have used in section 3.3 to show that Goodwin’s growth cyclemodel can be considered a special case of the Keynes—Wicksell prototypemodel. We also assume again

�� 0, in order to obtain a two-dimensional

dynamical system (in �, l) for this special case, which is independent of therest of the model. In this case we can easily solve (4.35) for the level of y andget from this equation:

y�s�(�� t�)� i

�(r��

�)� i

�� n

(s�� i

�)(1� u)� i

�/y�

�(s�� i

�)(1� u

�)� i

�/y�

(s�� i

�)(1� u)� i

�/y�

�(s�� i

�)(1� u

�)y�� i

�(s�� i

�)(1� u)y�� i

�

y��N

D· y�. (4.36)

For the denominatorD of this expression we can have one of the situationsdepicted in figure 4.1 (cases 1, 2a, and 2b: (s

�� i

�), case 3: (s

�� i

�)) if

u�� (0, 1) holds. Due to the last expression in (4.36), we always have

D� � i�at u� 1. The denominator D is thus positive in a neighborhood

of the steady state value u�only in case 1 where we have s

�� i

�� 0 and

sufficiently large relative to the size of i�. The benchmark case between

cases 2 and 3, finally, is given by a horizontal line (s�� i

�� 0) that passes

through�i�. Figure 4.1�� implies for the shape of the function y(u) the two

(four) possibilities� displayed in figures 4.2a�� and 4.2b.��Furthermore, the rate of profit �(u)� y(u)(1� u)� � is a decreasing

�� "will indeed be negative as long as y

�,y� � 0 and i

�/y�� h

�/h

�holds, i.e., for all i

�that are

sufficiently small.�� Note in figure 4.1 that the numerator (and the denominator) is negative at (around) the

steady state value of u in the cases 2a, 2b, and 3.� Note that, in the cases 3 and 2b, the delimiting vertical line at A

�,A

��must run instead

through the points A,A

�%, respectively. This enlarges the relevant phase space in an

obvious way.�� The sign of the derivative �� only becomes important when the dynamic law for relative

factor proportions l is considered, see the next section.�� y(u)� 0 as u�� in figure 4.2b.


Figure 4.1 The denominator in the effective demand function (4.36)

function of the wage share u, and thus of an expected type, in cases 2 and 3.This is obvious for cases 2a and 2b and follows for case 3 from

y(u)(1� u)� ��i�� (i

�� s

�)(1� u

�)y�

i�/(1� u)� (i

�� s

�)y�y�� ,

since the numerator of this expression is positive in case 3.Summarizing, we thus have, in the present context, y

�� 0, y� � 0, and

��(u)� 0 in these two cases, while case 1 behaves perversely insofar as wethen have a positive impact of the share of wages u on output y as well as onthe rate of profit � (by a similar reasoning as in case 3 just considered). It isto be expected that we will get a strange type of dynamical behavior in thislatter case.Finally, we make three further points. Firstly, we note that �(u) will be

constant in the case i�� 0, which should be considered a problematic

situation. Secondly, we see that in the case i�� 0 we have only the capacity

effect operating in the investment function, which nevertheless gives ameaningful situation (of type 1, 2a, or 2b).Thirdly, we observe that �

�will be positive in the present special case if


Figure 4.2a Effective demand in case 1 (�� 0) and case 3 (�� 0): a too weakcapacity effect (case 1) or s

�� i

�(case 3)

the data of the model fulfill s�(y�� t�)� n� 0. The steady-state value

of ��is then determined by s

�(y��

�y�/x� t

�)� n, which implies

that �� y��

�l�must be positive as well (n� 0). Due to �

��

y�(1� u�)� �, we then get u

�� 1 (and u

��

�/x� 0). This justifies our

assumption u�� (0, 1), made in figures 4.1 and 4.2. Note also that since r

�should be positive, one has to assume that �

��

�� n is positive, which is

justified for �sufficiently close to n, the natural rate of growth.

Putting the comparative-static discussion of this section into perspec-tive, one may be inclined to relate the results here presented to the dis-cussion on profit-led and wage-led growth as defined and investigated inparticular in various contributions to the volume edited by Epstein andGintis (1995). However we believe that, before this literature may becompared and considered, the present stage of our modeling of Keynesianmonetary growth must first be developed further into the direction of theworking model of chapter 6 and beyond, where the comparative-staticresults of the present section reappear in a robust dynamic, but much lessclear-cut and visible form. Several contributions in the volume by Epsteinand Gintis (1995) on structural macrodynamics (see in particular thepapers by Gordon) can, indeed, be usefully contrasted with the result we


Figure 4.2b Effective demand in cases 2a and 2b (�� 0): a strong capacity effect(relative to s

�� i

�� 0)

will obtain on the basis of the present section in the dynamical analysesthat follow, in particular concerning the mixed empirical results reportedby various contributors to the aforementioned volume, on wage-led vs.profit-led growth, on the full employment profit squeeze, the investment/savings relationship, and more. Structural macroeconomics, which at-tempts to integrate the insights of Marx, Kalecki, and Keynes for the short,the medium, and long run�� and which stresses the medium- and long-runaspects of macroeconomic interdependence, surely represents a decisivestep forward in the understanding of economic evolution and in thearticulation of a new macropolicy for sustainable economic growth, withwhich the results obtained in this book on integrated Keynesian dynamicsof traditional type must be compared and evaluated.��

�� See here also Dutt (1984, 1990, 1992) and Taylor (1983, 1984, 1991).�� See Flaschel (1998a—d) and Chiarella et al. (1999), who extend the working model of this

book to allow for an evaluation of such approaches to effective demand, capacityutilization, and the conflict about income distribution, and Chiarella and Flaschel(1998c—f ), who pursue such topics for the case of open economies.


4.3 Growth cycle implications

This section repeats the analysis of section 3.3 of the preceding chapter,now for the Keynesian model type. It shows, in particular, that a muchmore varied outcome for the Goodwin (1967) profit squeeze mechanismoccurs in the Keynesian version of real growth, which in one case bears noresemblance to the overshooting mechanism of the original growth cyclemodel.��The construction of a real subsector of the model which is dynamically

independent of the rest of the model is performed in the same way as in thegrowth cycle subcase of chapter 3. The assumptions r� r

�(the ‘‘liquidity

trap’’ at the steady-state rate of interest r�) and � �

�� n (infinitely fast

regressive expectations) imply again that investment per unit of capital willdepend on the wage share u��/x. However, now it also depends on therate of capacity utilization U� y/y�. The further assumption � 1 re-moves the Rose effect (the influence of the theory of inflation on thedynamics of real wages) from the dynamics of the real sector.Yet, we now have Keynesian demand problems, and therefore a Good-

win-type dynamical system of the kind

u��(y(u)/(xl)� 1), u��/x, (4.37)

l1 � n� s�(�(u)� t�), l�L/K, (4.38)

with �(u)� y(u)(1� u)� �. Where, y is no longer a constant (equal to y�),but determined by equation (4.36). Note that we no longer have V1 �� l1for the rate of employment V� l/l, since l� y/x is no longer a givenmagnitude but dependent on y, which in turn is a function of u. The use ofGoodwin’s variables u and V is therefore no longer a straightforwardmatter. Since the original Goodwin growth cycle is structurally unstable,wemay also expect that it will be changed into various subcases, dependingon whether case 1, 2, or 3 of the preceding section is considered asdetermining the behavior of y(u) and �(u).The Jacobian of the dynamical system (4.37)—(4.38) at the steady state is

given by

J��y�(u)/(xl)u ��

�y(u)/(xl�)u

��

�s��(u)l 0 � . (4.39)

�

For the three cases considered in the preceding section we thereby get thefollowing for the behavior of the dynamical system (4.37)—(4.38) near thesteady state.

�� See also Franke and Asada (1993) for a synthesis of Keynes—Goodwin type.


Proposition 4.2: The dynamical system (4.37)—(4.38) is describedlocally by saddlepath dynamics in case 1 (y�� 0,�� 0), by a stable nodeor focus in the cases 2a and 2b (y�� 0, �� 0), and by an unstable node orfocus in the case 3 (y� � 0, �� 0).

Proof: Evaluating the above Jacobian in these three situationsgives:

case 1: J��

� 0 � , cases 2a and 2b: J��

� 0 �,case 3: J��

� �

� 0 � .These three representations immediately imply the three assertions of theproposition by means of the usual characterization of local phase plots viathe signs of the determinant and the trace of the Jacobian.��

An appropriate application of Olech’s theorem (see Flaschel 1984 fordetails) should here imply that the last two stability characterizations alsohold in the large, i.e., in the positive orthant R

(.

The following proposition shows that cases 2 and 3 give rise to aGoodwin type dynamical behavior if the wage adjustment parameter ischosen sufficiently small.

Proposition 4.3: Cases 2 and 3 will exhibit cyclical dynamicalbehavior if and only if

�� 4

s��(u)u

(y�(u)u/y)�(� 0).

Proof: A straightforward implication of the calculation of thediscriminant�: 4�� (trace J)�� 4det J of the Jacobian of the dynamics atthe steady state, which is negative, and thus gives rise to complex roots,when the stated condition is fulfilled.�

The phase diagrams of the three types of dynamical behavior just consider-ed are displayed in figures 4.3a—c (see figures 4.1 and 4.2 in the precedingsection with respect to the boundary values A

#, i� 1, 2, 3).�

� See Flaschel (1993, ch. 4).� Note that in figure 4.3c case 2 will switch to case 3 (and vice versa) through an appropriate

change in the parameter i�in a continuous fashion and that the dynamicswill pass through

the Goodwin center type dynamics in this situation.


Figure 4.3a Case 1: the ‘‘paradise’’ case (y�(u),��(u)� 0): a saddlepoint

Figure 4.3b Case 2: the ‘‘orthodox’’ case (y�(u),��(u)� 0): stable node or focus


Figure 4.3c Case 3: the ‘‘mixed’’ case (y�(u)� 0, ��(u)� 0): unstable node or focus

We note here without proof (see section 4.4 for details) that the masspurchasing power argument is valid in the cases 1, 2a, and 2b in the sensethat an increase in the real wage will, ceteris paribus, increase excessdemand �y� y� y� (i

�� s

�)(1� u)y� i

�y/y�� const., which leads to

an increase in effective demand y in the first case and to a decrease of it incases 2a and 2b. The mass purchasing power argument is therefore onlysupported by case 1, when the goods-market reaction is taken into account.In case 3, finally, we have that the cost effect of real wage increasesoutweighs the immediate mass purchasing power effect, so that there isexcess supply created initially. Taking again into account the goods-market reaction to this disequilibrium situation, however, here neverthe-less gives that effective demand is increased by the real wage increase.It is possible to tailor further the dynamical system in the stable case 2 so

that its trajectories stay within a certain corridor around the steady state.To this end, assume, as in section 3.4, that the adjustment speed of moneywages with respect to labor market disequilibrium is of the nonlinear typedisplayed in figure 4.4.The value of l which together with y(u) satisfies l/l� y(u)/(xl)�V

�or

�V%is given by l

�� y(u)/(xV

�) or l

%� y(u)/(xV

%). Along these curves the

rate of change uwill be�� or��, respectively.We thereby get for case 2


Figure 4.4 The parameter ��(V ) of the wage adjustment function �

�(V )(V� 1)

the restricted phase diagram displayed in figure 4.5 since V�� 1 and

V%� 1, due to our normalization of the value of V� guaranteeing that the

employment rate will stay between 0�V�and V

%�V

��.��

The locally asymptotically stable situation of case 2 can thereby easily beextended to an economically meaningful domain around the level ‘‘1’’ ofthe ‘‘natural’’ rate of employment. A similar construction is not possible inthe other two cases. However, these will be further discussed in the nextsection, when the Rose real-wage mechanism is added to the dynamics.A number of other (unsatisfactory) features of the model should be

pointed out. Firstly, it still lacks a mechanism which keeps the evolution ofthe share of wages u below 1. Secondly, we note that all three curves u� � 0,l�, and l

%are increasing in case 3, i.e., the horizontal arrows of figure 4.5 are

then pointing outside instead of to the inside of the compact domain

�� Note that y(u)��(� 0) for u�A�(��). The shaded area depicted in figure 4.5

therefore always exists, but it may be bounded to the right by a value of u that is larger than1. In order to obtain a value smaller than 1 one has to assume in addition

y(1)/x� y(u�)/(xV

%), i.e.,

y(1)

y(u�)� 1�

(s�� i

�)(1� u

�)y�

i�

�1

V%

,

i.e., V%sufficiently close to 1.


Figure 4.5 A region of global stability for case 2

depicted there. The instability of case 3 is therefore significantly increased ifwages become more and more flexible the further the rate of employmentdeparts from its steady-state value 1, leading here to a collapse of thedynamics in finite time.��Let us finally consider here cases 1—3 from the perspective of ultra short-

run goods-market disequilibria and possible adjustment problems result-ing from them. The IS-equilibria we have considered above have beenderived from the goods-market equilibrium condition y� uy�

(1� s�)(1� u)y� i

�(1� u)y� i

�y/y�� const.� y, where const. in par-

ticular includes t�� . For the excess demand function we get from thisexpression, �y� y� y� (i

�� s

�)(1� u)y� i

�y/y�� const., and thus

for the partial derivative of this function with respect to output per unit ofcapital y we have

�y�� (i

�� s

�)(1� u)� i

�/y�

� 0, (case 1),

� 0, (cases 2a, 2b, 3)

�� InFlaschel (1993, pp.166ff.) case 3 (with i�� 0) ismade to have a viable dynamical behavior

via an appropriately delayed type ofmarkup pricing and the assumption that �will be less

than 1 for u close to 1.


(see figures 4.3a—c).From the perspective of simple textbook analysis and its dynamic multi-

plier story, one might stipulate�

y� ��(�y)��

�(y(y)� y),�

�� 0,

and thus obtain stability in case 1 and instability in cases 2 and 3. TheIS-equilibria of cases 2 and 3may therefore be rejected as tool of analysis aswe have used them here.The excess demand situation described above also helps to explain in

more depth why we had y�(u)� 0 in cases 1 and 3 and y�(u)� 0 in case 2.

Case 1: The aggregate demand function we have in this case isgiven by

y� y� [(i�� s

�)(1� u)� i

�/y�]y� const.,

�

�

�

which implies that a rising share of wages uwill give rise to a higher level ofaggregate demand for any given value of y, i.e., u � (y given) y � . . . y � .Excess demand is consequently created (the mass purchasing power argu-ment) if IS-equilibrium prevailed initially. A restoration of IS-equilibriumfurthermore requires a higher level of output, since we have in this casey�� 1. Taken together we therefore here obtain y�(u)� 0.

Case 2: The aggregate demand function in this case has the form

y� y� [(i�� s

�)(1� u)� i


�

(

�

which again implies that a rising share of wages u will give rise to a higherlevel of aggregate demand for any given value of y, i.e., u � (ygiven) y � . . . y �. Excess demand is consequently again created (themass purchasing power argument) if IS-equilibrium prevailed initially. Arestoration of IS-equilibrium now, however, requires a lower level ofoutput since we have in this case y

�� 1, so that excess demand is only

brought back to zero by a decrease in aggregate output (�y�� 0). Taken

together we therefore here obtain y�(u)� 0.

Case 3: The aggregate demand function in this final case has theform

� K of a given magnitude here.


y� y� [(i�� s

�)(1� u)� i


(

(

�

which now implies that a rising share of wages u will give rise to a lowerlevel of aggregate demand (real wage cost argument) for any given value ofy, i.e., u � (y given) y � . . . y � . Excess supply is here consequentlycreated if IS-equilibriumprevailed initially. A restoration of IS-equilibriumthen requires a higher level of output since we have in this case y

�� 1. This

excess supply is brought back to zero by an increase in aggregate output(�y

�� 0). Taken together we therefore here obtain y�(u)� 0.

In sum, we thus have that increases in real wage �(� ux) have an initialmass purchasing power effect in cases 1 and 2, since the extra consumptionthrough wage-earners’ additional income exceeds the reduction in capital-ists’ consumption and investment caused by this real wage increase (for agiven y). Output ymust then be increased in case 1 (and decreased in case 2)in order to eliminate again the demand gap caused by such a real-wageincrease. Only in case 1 does the argument of those who favor massconsumption therefore hold, that real-wage increases will lead to moreoutput rather than less, while there only seems to be increased (excess)demand in case 2 (as long as y is held constant), a situationwhich, however,must indeed lead to output shrinkage eventually when goods marketequilibrium is restored.Case 3, finally, does not support the mass consumption argument right

from the start, since real-wage increases here lead to a drop in aggregatedemand (y given), due to a strong reduction in investment demand. How-ever, since the marginal propensity to spend out of income is larger than 1(again due to investment demand), the resulting excess supply is removedby an increase in economic activity. Strong investment demand responsesto increasing output therefore here eliminate again the negative demandgap caused by the dominant cost effect of real-wage increases.But how do we cope with the situation that the dynamic multiplier,

which would be expected to lead us from old to new equilibria, is unstablein the last two cases?Following Benassy (1986a),�� we can first of all make the situation of

instability more manifest, by assuming, as does Benassy, that investmentdoes not depend on actual demand, but rather on expected demand, whichadjusts to actual demand with some lag. The resulting situation of anIS—LM equilibriummodel with delayed output expectations of investors is

�� See appendix 1 to this chapter for the presentation of a model of this type.


here described by the two equations (y*� expected demand per unit ofcapital)

y�� y� uy� (1� s�)((1� u)y� � � t�)� �� t�

� i�((1� u)y*� �� r

��

�)� i

�(y*/y�� 1)� n� y,

y� *� ��(y� y*).

Inserting the first equation (solved for y) into the second one then gives riseto [y� i

�( · )� i

�( · )� const.)/(s

�(1� u))]

y� *��

i�( · )� i

�( · )� s

�(1� u)y*� const.

s�(1� u)

��

[(i�� s

�)(1� u)� i

�/y�]y*� const.

s�(1� u)

,

which describes a stable adjustment process iff (i�� s

�)(1� u)� i

�/y�� 0

holds (iff case 1 holds). Hence, one may regard cases 2 and 3 as providingthe unstable situation of the Benassy IS—LM model, which may lead toglobal stability when an appropriate wage adjustment mechanism is addedto it as in Benassy (1986a).The first solution to the observed multiplier instability of cases 2 and 3

may therefore exist in an extension of the dynamics of this section bymeansof the third dynamical law

y� *��(y� y*). (4.40)

Our conjecture on the resulting three-dimensional dynamics is that thiswill increase the instability of case 3, and overthrow the stability of case 2, if��becomes sufficiently large. This implies that these dynamics must be

studied further, when more is known about constraining factors, which arecapable of ensuring the global stability of this extended dynamical system.The following sections will explore some possibilities for such global (oreven local) stabilizers from a two-dimensional point of view. Due to theincrease in the dynamical dimension that is caused by (4.40), computersimulations may, however, be essential in order to study the global behav-ior of this Benassy-like extension properly.Are there possibilities other than (4.40) which allow us to maintain the

use of IS-(LM-)equilibria when treating the system (4.37)—(4.38)?A first possibility (in fact now the orthodox position) would assume

myopic perfect foresight instead of the adaptive mechanism (4.40). Thisapproach is usually informally justified by the assumed reaction of agentsto the cumulative instability inherent in the expectational process (4.40).Assuming y� y* would therefore be very natural from such a point of


view, which then would apply the jump-variable technique in order toavoid cumulative instability.Another approach, the one we shall favor here, is provided by de la

Grandville (1986), who shows that stable and unstable IS-(LM-)equilibriacannot be characterized by the slopes of IS and LM curves alone: ‘‘For onesuch system and one inventory policy, the trajectories of output andinterest rate can either be stable or unstable’’ (p.31). Therefore, even with aspecified inventory policy (which is not considered in the case of the simpledynamic multiplier story), there is no clear-cut stability result to be ex-pected from the situation given in cases 2 and 3. This being so, oneapproach is to model the inventory adjustment process in its full detailsand analyze its various stability outcomes completely. Alternatively, onemay just stick to the use of IS-(LM-)equilibrium as a low-dimensionalrepresentation of a much more complicated, and rarely investigated, dy-namical system and accept that there does not exist a stable IS-(LM-)slopeconfiguration per se which really justifies the use of IS-equilibrium con-cepts. In this chapter we shall take this latter position, while the inventoryadjustment process is integrated into the dynamics of the Keynesian modelin chapter 6, leading to the central workingmodel of Keynesian type of thisbook.If, however, one insists that the IS-equilibrium should be stable in the

straightforward way of the dynamic multiplier story, there may still existfurther possibilities to improve the situation given by cases 2a, 2b, and 3,e.g., (i) by introducing a positive savings propensity of workers s

�� 0 and

the like,�� (ii) by allowing for interest rate flexibility (see section 4.5), or, (iii)by (as already described) using the Benassy modification of IS-equilibriumwith a sluggish adjustment of the income concept used in investmentbehavior to have a stable multiplier combined with an (in general de-stabilizing) expectations-correcting mechanism.We are aware at this point that future investigation has to improve and

modify the views here adopted. This task will be undertaken in chapter 6,where there will no longer arise the necessity to distinguish between thecases 1, 2a, 2b, and 3 of this section.

4.4 Employment cycle extensions

The three outcomes of the preceding section are here further analyzed byincluding again the more complicated real-wage mechanism of the Rose(1967) employment cycle, in which the dynamics of the real wage alsodepends on the state of the market for goods. We shall find that there is a

�� In such a case �y�� (i

�� s

�)(1� u)� i

�/y�� s

�u� 0 becomes more likely.


trade-off between the stabilizing/destabilizing effects of wage—price flexibil-ity depending upon the sign of y�(u). Such a scenario will again allow forRose’s (1967) nonlinear limit cycle analysis where, however, the stabilizingand destabilizing factors may now interchange their roles.We therefore extend the dynamic feedback possibilities of the model as

in section 3.4 by allowing for �

� 1 while keeping all other assumptions asin section 4.3. The dynamical system for u, l then reads (recall �(u)�y(u)(1� u)� �):

u� [(1� �)��(y(u)/(xl)� 1)� (

�� 1)�

�(y(u)/y�� 1)], (4.41)

l1 � n� s�(�(u)� t�), [g� t�]. (4.42)

From equation (4.42) we can determine the steady-state value of u (vial1 � 0) as given by y(u

�)(1� u

�)� � � t�� n/s

�. With respect to this value

of u we have, by assumption, r��(u

�)�

�� n, and therefore

n� s( · )� i�( · )� i

�( · )� n, with i

�( · )� 0, i.e., i

�( · )� 0, which implies

y(u�)� y�, i.e., we have full capacity utilization at the steady-state value of

the share of wages u. By equation (4.41) we then get for the steady state (viau� 0): y�/(xl

�)� 1 or l

�� y�/x, as in the Goodwin subcase considered in

the previous section.The above extension of the Goodwin case implies for the entry J

��of the

Jacobian J of equation (4.39) in the preceding section

J��

� u [(1� �)��/(xl)� (

�� 1)�

�/y�]y�(u),

while all other elements in the matrix J remain unchanged. This immedi-ately implies the following:

Proposition 4.4: (1) Increasing price flexibility ��destabilizes case 2

(y�,�� 0) and stabilizes case 3 (y�� 0, �� 0) of the preceding section asfar as the local asymptotic stability of the steady state is concerned. Theseswitches in stability will occur by way of Hopf bifurcations;� (2) Thereverse effect holds for increases in wage flexibility �

�. These switches in

stability will again occur by way of Hopf bifurcations.

In sum, we get in this Keynesian context that either increasing priceflexibility or increasing wage flexibility may be stabilizing, but not both.Which one will be stabilizing depends (if i

�is sufficiently large so that case 1

can be excluded) on the sign of (i�� s

�), and therefore on the relative

sensitivity of investment and savings to changes in the real wage. If equilib-

� Since the determinant of the Jacobian is always positive in the given situation.


rium output y falls (rises) with a rising real wage, the stabilizing role falls on��(��).

In order to obtain phase portraits for cases 2 and 3� we calculate theu� � 0 and l� � 0 isocline of these two cases first. For the l� � 0 isocline wehave, as in section 4.3, that �(u

�)� t�� n/s

�, which determines a vertical

curve in (u, l)-space. And for u� � 0, we obtain by (4.41) l� (1/x)/((1� q)/y(u)� q/y�), where we set q� (1�

�)��/(1�

�)��.

The u� � 0 isocline is well defined as long asU� y(u)/y�� 1� 1/q (notethat q� 0) holds true, since the denominator of the above fraction is thenpositive (and only then). This domain includes the steady-state valueU

�� 1, but the interval to the left ofU

�that it includes becomes small if �

�becomes large relative to �

�. The derivative of this isocline reads l�

(u)� (1� q)y�(u)/([(1� q)/y(u)� q/y�]�xy�), i.e. l(u) is strictly decreasingin case 2 (y�� 0) if q� 1 holds and strictly increasing if q� 1. For q� 1,the isocline will be strictly horizontal. Case 3 (y� � 0) will give rise to resultsthat are exactly opposite to those of case 2.For q� 1, we have 1� 1/q� 0, i.e., the l� � 0 isocline is well defined for

all u for which y(u) is well defined (see the figures in section 4.3). Further-more, its slope is of the same type as that of the l� � 0 isocline in section 4.3,i.e., case 2 and case 3 have the same phase portraits as in the Goodwin caseas long as �

�is so small that q� (1�

�)��/(1�

�)��

� 1 holds true.As q approaches 1 (from below) the u� � 0 isoclines in these figures will

become flatter and flatter and are strictly horizontal lines finally for q� 1.This reestablishes the closed-orbit structure of the original Goodwinmodel, since the model then reduces to

u� (1� �)��y(u)�

1

xl�

1

y�� , (4.43)

l1 � n� s�y(u)(1� u))� �� t�, (4.44)

which can be treated in the sameway as the Goodwin-like versions we haveestablished for the Keynes—Wicksell prototype model in chapter 3. Inaddition to proposition 4.4, we can now state that case 2 remains of thestable type we considered in the preceding section and that case 3 remainsof the unstable type as long as q� 1 holds, and that both cases undergo abifurcation at q� 1 (passing the Goodwin closed-orbit structure at thisvalue of q) and then become explosive for q� 1 in case 2 and stable forq� 1 in case 3.Case 2 therefore switches from stability to instability at q� 1 because of

� Case 1, the paradise case, is dismissed here because it looks too unlikely from an empiricalpoint of view.


Figure 4.6 Instability for case 2 via the Rose effect (y�� 0, q� 1,��

��

��)

the Rose (or ��) effect of price formation on real wages, giving rise then to a

phase portrait of the type displayed in figure 4.6.By contrast, case 3 switches from instability to stability by passing

through the Goodwin center case at q� 1, now giving rise to a phaseportrait of the type shown in figure 4.7 when �

�is further increased.

We claim, but do not prove here, that this case should also be asymptoti-cally stable from a global point of view, that is, each trajectory which startsbetween u� and A

(at not a too high level of l) should converge to the

steady state (u�, l�).

Let us now also consider the unstable cases, case 2 with q� 1 and case 3with q� 1, from a global point of view.Wewant to see to what extent thesecases may be stabilized far off the steady state by either a nonlinearity inwage adjustment or in price adjustment of the type we have considered inthe Rose section of the Keynes—Wicksell prototype.Case 2 (q� 1 at the steady state): Assume, as in the preceding section,

that wage adjustment becomes infinitely fast as the rate of employmentV� y/(xl) approaches the level 0�V

�� 1 from above and V

%� 1 from

below. This nonlinearPhillips curvemechanism is displayed in figure 4.8.�

� Note that we now employ a nonlinear adjustment function ��(V ) in the place of the linear

adjustment function ��· (V� 1). Note also that q(1)� ((1�

�)/(1�

�)) �

�/��(1)� 1 is

assumed here.


Figure 4.7 Stability for case 3 via the Rose effect (y�� 0, q� 1)

Figure 4.8 The nonlinear Phillips-curve mechanism once again


By contrast, the price adjustment speed is assumed as constant in thepresent situation and of such a size that q(1)� ((1�

�)/(1�

�)) �

�/

��(1)� 1 (i.e., �

�� ((1�

�/(1�

�))�

�(1)) holds. In place of the above

u� � 0 isocline we would then obtain

l�y(u)/x

��

1� �

1� �

��y(u)

y�� 1��

,

which is well defined as long as y(u) is (see section 4.3), since V�� 0 has

been assumed. Furthermore, l%(u)� l(u)� l

�(u), where l

�(u)� y(u)/(xV

�),

l%(u)� y(u)/(xV

%) as in the preceding section. The last phase diagram for

case 2 is therefore modified by this nonlinearity in wage adjustment behav-ior in the way shown in figure 4.9, since the denominator of the u� � 0isocline is restricted to the interval [V

�,V

%], while y(u)�� for u�A

�(from above), and y(u)� 0 for u��. The existence of the shadedcompact domain in figure 4.9 allows in principle for the application of thePoincare—Bendixson theorem. Note that this domain can extend to theright of u� 1, and thus needs in this case further restrictions to guaranteethat the share of wages u stays below u� 1. This can be done, for example,bymeans of an appropriate choice ofV

�and l

�. The result we have obtained

in this way is summarized by the following.

Proposition 4.5: All trajectories which start in the shaded domaindepicted in figure 4.9 approach a limit cycle contained in this domain thatcontains the steady-state u

�, l�in its interior (or are equal to such a limit

cycle which may not be uniquely determined).

Wage flexibility may therefore be the appropriate globally stabilizingfactor in the case we have called the orthodox one (y�(u)� 0) for a fixed typeof price flexibility, that was chosen large enough to destabilize the steadystate locally.Case 3 (q� 1 at the steady state, y�(u)� 0, u � (0, 1)): In principle, price

flexibility should now be stabilizing if it increases sufficiently far off thesteady state (for any given degree of wage flexibility �

�), since it leads to

q� 1 far off the steady state.

The picture in figure 4.9 is obtained with u� � 1 if it is assumed that

y(1)/(xV�)� y(u

�)/(xV

%), i.e.,

y(1)

y(u�)� 1�

(s�� i

�)(1� u

�)y�

i�

�V�

V%

(� 1)

holds (recall that s�� i

�in this case), i.e., if V

�and V

%are sufficiently close to 1. Note here

that a construction as in figure 4.5 of the preceding section allows the use of the value 1 inthe place of V

�, which is a less-restrictive assumption.


Figure 4.9 Viability in the locally unstable case 2 (the real cycle, case 1)

Figure 4.10 A second Phillips-curve mechanism


Let us assume in analogy to case 2 above that the situation depicted infigure 4.10 holds for the price adjustment coefficient �

�. In this case we have

for the u� � 0 isocline

l�y(u)/x

1�1�

�1�

�

��(y(u)/y�)

��

,

which is only well defined as long as c��(1�

�)/(1�

�)��

�( · )

holds true (see figure 4.10 [note thatU�U�, u� u�, since y�(u)� 0 holds

in this case�]). It can be shown that the phase portrait that is implied bythis new situation is as depicted in figure 4.11.This situation will again allow for an application of the Poincare—

Bendixson theorem, now leading to the following.�

Proposition 4.6: All trajectories which start in an appropriatelychosen compact domain of figure 4.11 approach a limit cycle that iscontained in this domain.

We make a number of remarks in respect of the foregoing discussion.Firstly, the assumption of only the boundary behavior of the price level atU%is already sufficient for such a limit cycle result. Secondly, the unstable

(mixed) case 3 of Goodwin type is here stabilized by some sort of anti-Roseeffect. Thirdly, the fact that theU

%boundary is vertical is new and different

from the l%situation, but of course l

%and l

�refer to infinite wage flexibility,

which is of no help in the present situation.In sum, we therefore have the results that �

�flexibility works against �

�instability in case 2 (y�� 0), while �

�flexibility works against �

�instability

in case 3 (y�� 0). Thus, the kind of flexibility that is needed for the globalstability of the dynamical system depends on the i

�� s

�regime. Whatever

this particular regime may happen to be, one flexibility is always stabiliz-ing, while the other is then necessarily destabilizing.We make one final remark. Consider briefly case 1, to see what happens

Note that we now employ a nonlinear adjustment function ��(U) in the place of the linear

adjustment function ��· (U� 1).

� We assume that U� and U%are chosen such that U�� y(u�)/y�: U

%� y(u

%)/y� determine

values of u with 0� u�� u%� 1.

� Note here that the upward sloping line in figure 4.11 still needs to be justifiedwith respect toits existence and that trajectories are ‘‘pointing inwards horizontally’’ at the right handboundary of the depicted shaded domain. The application of the Poincare—Bendixsontheorem is therefore not straightforward in this situation.

This result is now based on local instability of the steady state caused by a sufficiently highwage flexibility, turned into global stability by an ever-increasing price flexibility far off thesteady state.


Figure 4.11 Viability in the locally unstable case 3 (the real cycle, case 2)

in this case when q passes through 1 and becomes larger than 1. Figure 4.12depicts the relevant phase diagrams. Here we see that the dynamicalbehavior in this case remains of a saddlepoint type even if prices becomevery flexible. This case therefore cannot be made asymptotically stable byincreasing wage or price flexibility.

4.5 Keynesian monetary growth: the basic case

In this section we now dispense with the assumption h��(r� r

�) main-

tained in the analyses of sections 4.3 and 4.4. We show that the addition of(sufficient) interest rate flexibility may, due to the Keynes effect, generate(local) asymptotic stability in the now integrated real andmonetary growthdynamics of this section, and may thus eliminate the Rose-type limit cycles.We shall, on the other hand, see that the potential for local Hopf bifurca-tions, for example, with respect to speeds of price adjustment, is significantin this three-dimensional extension of the two-dimensional dynamics ofpreceding sections. The interaction of real and monetary factors can there-fore still give rise to cyclical evolutions for a given value of the parameter h

�


Figure 4.12 The stability switch in case 1

if speeds of price adjustment become sufficiently high, and may thusexplain the joint occurrence of growth and fluctuations on the macro level.Note here that we still maintain the assumption �� (asymptoticallyrational expectations) in this section, so that the destabilizing role of theMundell effect remains excluded from the investigation of the dynamicalbehavior of the model.We have so far assumedwith respect tomoney demand that it is given by

m(y, r)� h�y� h

�(r�� r),m

�� h

�� 0,m

�� h

�� 0

(with h�� in the analysis of the last two sections). LM-equilibrium thus

gives rise now to the expression:

r� r��h�y�m

h�

(� r�if h

�� )

as the implied determination of the nominal rate of interest. Let us now goto the opposite extreme and consider values of the parameter h

�which are

sufficiently small (to be characterized below) that the rate r may nowbecome very sensitive to changes in the values of m or y.The investigations of the present section will all be local in nature

(concerning only a certain neighborhood of the steady state of the model).Proposition 4.1 of section 4.2 implies that, for any h

�chosen sufficiently

small, the partial derivatives of this proposition are all well defined andfulfill


y"� 0 iff s

�� i

�, y�� 0� (4.45)

for the function y(u,m) that is then implied by equation (4.35) and for aneighborhood of the steady state. We assume in this section that h

�is

chosen such that (4.45) holds true. It is also easy to show by means ofequation (4.35) that the function �(u,m)� y(u,m)(1� u)� � must have anegative derivative �

"(u)� y

"(u)(1� u)� y(u) for all combinations of the

s�, i�parameters, if h

�is sufficiently small, so that there is no longer a

possibility for case 1 as in sections 4.3 and 4.4.Since we have only added interest-rate flexibility to the analysis of the

preceding section (but still maintain the assumption of � held equal to�� n), we now have to consider the three-dimensional dynamical system

�� [(1� �)��X�� (

�� 1)�

�X�], (4.46)

l1 � n� s( · ), (4.47)

m�� [��X��

��X�] l1 , (4.48)

where

X�� l/l� 1, l� y/x,X� � y/y�� 1,s( · )� s

�(�� t�)�

�m,

and with � � y� ��l� y(1� u)� �, and u��/x given as discussedabove.After a little algebraic manipulation the dynamical system (4.46)—(4.48)

can be represented in the form

u� � (��X��

�X�)u, (4.49)

l� � (const.� s��

�m)l, (4.50)

m� �� [��X��

�X�� s

��

�m� const.]m, (4.51)

where ��,��,��,��are all positive constants.

In order to investigate the stability of the dynamical system (4.49)—(4.51)at the steady state (as determined in section 4.1), we have to consider theeigenvalue structure of its Jacobian

J��(��X�"

� ��X�")u ��

�X��u (��

�X��

� ��X��)u

�s��"l 0 �s

��l�

�l

�(�� X�"��

�X�"

� s��")m ��

�X��m �(��

�X��

��X�� s

��

��)m�(4.52)

� Note that this differs considerably from the y"characterizations given in sections 4.3 and

4.4. Note also that the final partial derivative y� (� 0 here) is not yet of importance in thissection. Its sign represents the Mundell effect of a rise of inflationary expectations on thelevel of effective demand, while y

�� 0 is an implication (or reformulation) of the Keynes

effect of conventional IS—LM models.


evaluated at the steady state u� u�, l� l

�,m�m

�. Since the parameter

�can be considered as small we shall neglect it in the following calculationsof stability criteria.�

Lemma 4.1: The determinant det J of the Jacobian J in equation(4.52) has the same sign as �y

�, that is to say its sign is completely

determined by the sign of the Keynes effect.

Proof: Subtracting an appropriatemultiple of the second row fromthe third row and then an appropriate multiple of the first from the thirdgives

det J� �$ $ $

$ 0 $

�(��X�"��

�X�")m ��

�X��m �(��

�X��

�X��)m�

� �$ $ $

$ 0 $

�X�"m 0 �X�

�m�

where is equal to ��

�/��(m/u) and where $ denotes unchanged

entries of the Jacobian J.This latter determinant in turn can be shown in the same way to be equal

to

��X�"u ��

�X��u ��

�X��u

�s��"l 0 �s

��l

�X�"m 0 �X�

�m ��

�X��u ��s

��"l �s

��l

�X�"m �X�

�m �

�� X��u ��s

�y"(1� u)l� s

�yl �s

�y�(1� u)l

�y"/y�m �y

�/y�m �

��ums

�yly�X�

�y�

�� cy�

where c is positive, since X�� y/(xl�) is always negative.�

Lemma 4.2: The three leading principal minors of the Jacobian Jin equation 4.52 read

J��

J��

J�

J�

J �� s

�l��my

�(1� u)X�

�

� Such an assumption is justified, for example, if s��

��holds.


J��

J��

J�

J�

J �� s

�yum(��

�/y��

�/(xl))y

�

J� �

J��

J��

J��

J�� s

�l��X��u�

",�"� y

"(1� u)� y� 0.

Proof:Given that J��

� 0, the only nontrivial calculation is for J�.

Adding a multiple of the last column of J�to its first one gives

J��

0 $

�s�y(� 1)m $ �� s

�ym(��

�X��

��X��)u.�

Proposition 4.7: The steady state of the dynamical system (4.49)—(4.51) is locally asymptotically stable if s

�� i

�and �


Proof: Due to y�� 0, �

"� 0, throughout, we have det J� 0 and

J�, J

� 0 (since X�

�� 0). A sufficiently small �

�will then also make

J�� 0. Furthermore, the trace of J is given by

(��y"/(xl)� ��

�y"/y�)u� (��

�y�/y��

�y�/(xl)� s

��)m,

where �� y

�(1� u). In the case s

�� i

�we know that y

"will be positive

while y�and �

�are always positive. Therefore, trace J is negative in all of

its components, apart from the first term, which will not be relevant for itssign if �

�is chosen sufficiently small. Consider, finally, b� (� trace

J)(J��J

�� J

)�det J. It is easy to show that b� 0 holds when �

�� 0 is

assumed. Thus, b� 0 for all ��sufficiently small. Thus the sufficient

conditions for local asymptotic stability of the Routh—Hurwitz theorem(see the appendix to section 1.8) all hold true here.�

Corollary 4.2: Proposition 4.7 also holds if the parameter ��is

chosen sufficiently large instead of ��being chosen sufficiently small.

Proof: Since det J does not depend on ��, while trace J and

J��J

��J

both depend linearly and positively on it, (recall y

"� 0 under

the given assumption s�� i

#), we can always choose a �

�sufficiently large

such that J�� 0, trace J� 0 and b� 0 will be fulfilled.�

The case s�� i

�can therefore always be stabilized by a sufficient degree of

wage inflexibility or price flexibility. Price flexibility (wage flexibility) in thiscase therefore leads to local economic stability (instability). Of course, sinceonly trace J, J

�and b can create stability problems in a fairly simple


fashion, there will also exist a variety of further cases where local asym-ptotic stability will hold true.If the parameter h

�in the money demand function is chosen sufficiently

small, we not only know that (4.45) must hold, but obtain also (see therelevant expressions in proposition 4.1) that y

�� 1, y

"� 0 as h

�ap-

proaches zero. Thus trace Jmust always be negative for parameters h�that

are sufficiently small. This result allows us to prove the following proposi-tion in respect of the case s

�� i

�.

Proposition 4.8:The assertion of proposition 4.7 will also hold truefor s

�� i

�(y"� 0) if the parameter h

�is chosen sufficiently small.

Proof: An obvious adaptation of the proof of proposition 4.7 andthe foregoing remarks of the sign of trace J for h

�sufficiently small.�

We thus end up with the result that the case of a small parameter h�always

favors ��flexibility and �

�inflexibility as carriers of local asymptotic

stability. Recall that in section 4.4 we saw that the case of a large (infinite)parameter h

�favored wage flexibility in one case (where price flexibility

was then destabilizing) and price flexibility in another case (where wageflexibility was destabilizing). The new asymmetry between wage and priceflexibility of this section seems to be due to the Keynes effect y

�� 0 (or

y�� 0), which was absent in the case h

�� (r� r

�) of section 4.4.

We also deduce from the above propositions that assuming that bothwages and prices will react to disequilibria in a sufficiently sluggishwaywillnot always lead to local asymptotic stability. The outcome rather willdepend on the relative size of �

�and �

�, though an increase in interest

sensitivity increased may overcome the possibility of instability.We now prove two propositions concerning the occurrence of Hopf

bifurcations in the dynamical system (4.49)—(4.51).

Proposition 4.9:Assume s�� i

�, so that y

"� 0. There is exactly one

value ��

� 0 for the parameter ��which separates asymptotically stable

steady states (��

�) from unstable ones (�

��

�). At ��

�aHopf bifurca-

tion occurs.

Proof:Due to det J� 0, the quadratic function b(��) provides the

determining expression (i.e., b(��)� 0) for the minimum value of ��

�, where

the stability of the steady state is lost for the first time. Furthermore, therecan be only one situation where b(�

�) crosses the positive part of the

horizontal axis twice, since we already know b(0)� 0. This situation isdepicted in figure 4.13.


Figure 4.13 Determination of the bifurcation parameter value ��

When either�trace J or J��J

��J

(which are linear functions of �

�)

are strictly decreasing, they must cut the horizontal axis between the values��and ��

�shown in figure 4.13, i.e., local asymptotic stability cannot be

reestablished then after ��is crossed. If, however, both �trace J and

J��J

��J

are strictly increasing,� they are given by the functional

forms

�trace J� a�� a

�,

J��J

��J

� b

��

� b�,

where all coefficients are strictly positive. But then

(� trace J)(J��J

��J

)� a

�b�� (a

�b�� b

�a�)�� a

�b�,

which gives a quadratic function that assumes its minimum at

��

��(a

�b�� b

�a�)

2a�b�

� 0, i.e., the situation depicted in figure 4.13 is then

not possible. Therefore, no reswitching towards stability is possible once��has been crossed.

� We neglect here the borderline case with slope 0.


The proof of a Hopf-bifurcation situation at ��is now a routine exercise

(see Benhabib andMiyao 1981 for details, in particular with respect to theuse of Orlando’s formula for the proof that the eigenvalues cross theimaginary axis at ��

�with a positive speed).�

Proposition 4.10: Assume s�� i

�, so that y

"� 0. There is exactly

one value ��

� 0 for the parameter ��which separates asymptotically

stable steady states (��

�) from unstable ones (�

��

�). At ��

�a Hopf

bifurcation occurs.

Proof: Completely analogous to the proof of proposition 4.9.�

We should point out here that similar results with respect to the parameter��are not so easily obtained and may possibly work in the reverse

direction.Finally in this section, we note that the above calculations give quite a

bit of information about the Jacobian J. In particular, they have shownthat (i) sign det J (and sign J

�) depends solely on the sign of the Keynes

effect y�, where we always have y

�� 0; (ii) sign J

depends solely on the

sign of the profitability effect, �"(� 0 always); (iii) on the other hand, sign

(trace J) depends on the signs of y�and y

"as well as on the relative sizes of

the parameters ��, ��(in their interactionwith y

");� (iv) sign J

�depends on

the sign of y�in interaction with the relative sizes of the parameters �

�,��;

and (v) the term b� (� trace J)(J�� J

��J

)�det J is more difficult to

judge and must be analyzed as in the proof of proposition 4.7. Should�trace J, J

�, J

�, and J

, however, all be positive, it suffices to check

J��J�J�and J

�J��J�for positivity, since these are the only elements in

det J which can then overturn the positivity of b.

4.6 Monetary and real factors in Keynesian cyclical growth dynamics

In sections 4.3 and 4.4 we analyzed the dynamics of the real sector of ourKeynesian prototypemodel. In section 4.5 we added interest-rate flexibilityto the dynamic analysis, thereby increasing the dimensions of the dynami-cal system being analyzed to three. In that section the feedback fromexpectations formation was switched off. Our aim in this section is toincorporate the expectations feedback mechanism and thus analyze thedynamics of the full four-dimensional model equations (4.30) to (4.33). Wefirst discuss the expectations mechanism and various important subcases.Then we analyze the pure monetary cycle (i.e. the m, � dynamics) inherent

� This is the Rose conflict between ��and �

�.


in the model.We then consider the real cycle when the investment functionhas a particular nonlinear form which we use as a mechanism to constrainthe dynamics in the situation of local instability. Finally in this section, weconsider the interaction of the real and monetary cycle using numericalsimulations.In this chapter we have shown that the results we obtained for the case of

the Keynes—Wicksell model type in chapter 3 are not completely over-thrown when this supply-side Keynesian model is modified to a demand-side version of the determination of the output of firms. On the contrary,the cyclical growth dynamics of the Keynes—Wicksell case reappears in theKeynesian case, though it is now embedded in a much more diversifiedstructure of possible stability scenarios. This overall result is perhaps nottoo surprising, since our changes to the Keynes—Wicksell model, thoughimportant from the viewpoint of proper theorizing, have been few. Thewage—price module therefore still seems to play a dominant role in thedetermination of the dynamics of the model.

4.6.1 The expectations mechanisms

In this subsection we consider the impact of expectations formation on thedynamics of the models of this chapter. We proceed as in section 3.7 of thepreceding chapter by considering the main important subcases. As theanalysis is analogous to that section we only give brief comments here.

Regressive expectations (��1� 0,��2 ��): This special case of our gen-eral expectation mechanism again does not modify the results of thethree-dimensional case where expectations have been assumed to be al-ways equal to the steady state rate of inflation.

Adaptive expectations (��2� 0,��1��): In the case of adaptive expecta-tions, the four-dimensional dynamics becomes fully interdependent, sincethe evolution of � now depends on �, l, and m and that of �, l, and m on �.The evolution of inflationary expectations � is in this case determined byequation (4.33) with �� 0. From this we calculate for the dependence of �on itself the expression

�� [��y�/y�� y�/(xl)]. (4.53)

This expression (�Jof the Jacobian of the extended dynamics) shows

that the model of section 4.5 can also be made locally unstable via theaddition of adaptive expectations by choosing the parameter �� sufficient-ly high, if the Mundell effect y� is normal (y�� 0). As is known from othermodels we thus also obtain here, under the assumption just made, the


result that adaptive expectations create (at least locally) explosive behaviorif they become sufficiently fast. However, this situation is now no longer asuniversal as in the Keynes—Wicksell case.

Myopic perfect foresight (��2� 0,��1��): The fact that trace J ap-proaches�� for �� in the just considered case of adaptive expecta-tions again indicates that the limit case �� , i.e. � � p, may be of aproblematic nature. In this case, the two Phillips-type adjustment mechan-isms (4.21) and (4.22) of our general framework reduce to

�� (y/(xl)� 1), (4.54)

��

�(y/y�� 1), (4.55)

where y is determined by equation (4.35) of section 4.2. This case thereforegives rise to two different and seemingly contradictory real wage dynamicsif

�� 0 and �

�� holds true unless labor-market disequilibriumV� 1

and goods-market ‘‘disequilibrium’’ are always proportional to each otherwith the proportionality factor ��

�/(�

� �). This implies that the utiliz-

ation rates of the two factors of production are strictly inversely related toeach other and therefore give rise to a ‘‘perverse’’ sort of Okun’s Law. In thecase

�� 0, by contrast, we always have full utilization of the capital stock

and therefore another pronounced departure from the validity of Okun’sLaw. We conclude that the case of strict myopic perfect foresight is againproblematic from an economic point of view.

Forward and backward looking expectations: From a formal point of viewthis case represents the summation of the case of adaptive and regressiveexpectations and it thus inherits the stability and instability features of itstwo limit cases we have just discussed. These now combined expectationmechanisms can also be represented by the relationship

�� [�p� (1� �)(�� n)��], � �

��

,

�� . (4.56)

This form states that a certain weighted average of the currently observedrate of inflation and of the future steady-state rate is the measure accordingto which the expected medium-run rate of inflation is changed in anadaptive fashion.Our discussion of the perfect foresight case suggests that �� 1 should

hold at all times so that medium-run expectations of inflation are nevergoverned by the short-run actual rate of inflation solely, but should alwayscontain some nonmyopic forward-looking component. By contrast, it is


perfectly legitimate to set the adjustment speed parameter �� equal to �,giving �� p� (1� �)(

�� n) as the rule for inflationary expectations.

Infinite speed of price adjustment (��): This case is considered in

detail in section 5.3.2 of chapter 5, where it is in particular shown that,augmented by smooth factor substitution, it represents the prototype caseof Keynesian dynamics of the macroeconomic literature that derives fromthe so-called neoclassical synthesis. We shall there also see that this casecan again be characterized as supply-side Keynesianism. In fact, the Key-nesianmodel with a perfectly flexible price level is basically identical to thesame limit case for the Keynes—Wicksell model where the price level andthe nominal rate of interest adjust aggregate demand to the predeterminedlevel of aggregate supply at each moment in time.

4.6.2 The pure monetary cycle

In this subsection, we assume on the basis of the above discussion that theparameter values �

�, ��, and �� are all positive and finite, and thus exclude

from consideration the one-sided limit cases we have just considered. Wehere also assume

��

�� n for reasons of simplicity.

In order to derive the pure form of the monetary cycle we shall againmake use of the following two sets of assumptions:

∑ ��

� 0, �� 1: The real wage is thereby made a constant of the model

and it is set equal to its steady-state value in addition.∑K1 � n�L1 : The labor intensity l�L/K thus is a constant in the follow-ing model and it is set equal to its steady state value l in addition.

Both sets of assumptions can be justified in the usual way by stating thatthe intent of the present investigation is confined to some pure sort ofmedium-run analysis. They here simply serve to reduce the dimension ofthe dynamical system (4.30)—(4.33) by two to two (in the variablesm and �).The resulting dynamical system reads�

m� �� n��

�(y/y�� 1), (4.57)

�� (y/y�� 1)� ��(�

� n� �), (4.58)

where the output—capital ratio y is given by

s�(y(1� u

�)� � � t�)� n� nm� i

�(y(1� u

�)� �

� (r�� (h

�y�m)/h

�)� �)

� i�(y/y�� 1). (4.59)

� Note that the output—capital ratio and the rate of profit are not constant in the presentcontext. Here we set u

��

�/x and

�� 0 for simplicity.


As in sections 3.4 and 3.6 of the preceding chapter, the following investiga-tion makes use of a nonlinear i

�component of the investment function i( · ),

here of the type

i�(�� r��)� c

�· tanh(i

�(�� r��)/c

�),� � y(1� u

�), c

�� 0.(4.60)

This function has slope i�at the origin and its range is limited to the

interval (� c�, c

�) as can be easily checked.With respect to this investment

function one gets for the partial derivatives of the function y(m,�) implicitlydefined by the goods-market equilibrium condition, i.e., equation (4.59),

y��

n� i��( · )/h

�(s�� i�

�( · ))(1� u

�)� i�

�( · )h

�/h

�� i

�/y�,

y��i��( · )

(s�� i�

�( · ))(1� u

�)� i�

�( · )h

�/h

�� i

�/y�.

These partial derivatives are well defined and nonnegative if the conditionss�(1� u

�)� i

�/y�� 0, h

�/h

�� (1� u

�)� 0 hold, since the denominator in

the above fractions is then always positive (and larger than s�(1� u

�)� i

�/

y�). In this case we always have a normal Keynes effect y�

� 0 as well as anormal Mundell effect y�� 0 associated with the considered dynamics.The steady state of the dynamical system (4.57)—(4.58) is as determined in

section 4.1, viz. m�� h

�y�, �

��

�� n� 0, since we assume

�� n. Its

Jacobian at the steady state is given by

J��

�y�/y�m �(1� �

�y�/y�)m

�� y�/y� ��

�y�/y�� . (4.61)

The sign of the determinant of this Jacobian is easily shown to equal thesign of y

�and is thus positive, while the trace of J is given by �

�/

y�(��y�� y�m)��. For �� 0 we therefore get a positive determinant

and a negative trace of the matrix J and thus local asymptotic stability ofthe steady state. It is easily shown, furthermore, that the dynamical systemundergoes a Hopf bifurcation when the parameter �� is chosen sufficientlylarge, due to the dominance of theMundell effect y� that then comes about.The following construction of conditions that imply the validity of the

Poincare—Bendixson theorem gives rise, however, to a situation that ismuch more general than that of a Hopf (limit) cycle (or that of a Hopfclosed-orbit structure) at some intermediate value of the parameter ��. Tothis end we have first to calculate the slopes of the isoclines for the (�,m)phase diagram of the above dynamical system. These are given by:


Figure 4.14 The pure monetary cycle

for m� � 0,m�(�)��

y� �1

��/y�

y�

� 0,

and

for �� 0,m�(�)��

y��

�� /y�

y�

� 0.

These isoclines give rise to the phase diagram of figure 4.14 for this puremonetary cycle model. The invariant box shown in this diagram can beobtained by choosing the parameter �� appropriately large such that the�� 0 isocline cuts the horizontal axis just once (to the left of the steady-state value �

�� 0). If this is given, the rectangle shown can be constructed

by following the sequence of points A,B,C,D. Note here that the condi-tions on the parameter ��may be such that the slope of the �� 0 isoclineis positive throughout, in which case there is a stable steady state and thus


Figure 4.15 A numerical example for the pure monetary cycle

no compelling reason for the existence of a limit cycle in the above diagram.In the opposite case (where the situation depicted in figure 4.14 holds) theexistence of at least one limit cycle, and the proposition that all trajectoriesin the above domain are attracted by one such cycle (or identical to it), arean immediate consequence of the Poincare—Bendixson theorem. The as-sumptions made indicate, however, that the scope for the application ofthis theorem in the present context may be small.Figure 4.15 shows a numerical simulation of this limit growth cycle

which is based on the parameter restrictions we have discussed above. Theparameter values of this simulation are shown in table 4.1.This collection of plots shows a monetary (limit) cycle similar to the one

we depicted in section 3.6, and also shows the nonlinearity of the invest-ment function employed in the generation of this cycle. The lower twodiagrams add some time-series plots to this numerical example of the m,�dynamics with an interesting pattern for the real and the nominal rate ofinterest. Note here that the amplitude of the fluctuations of the rate ofinflation is fairly large in this pure monetary cycle.


Table 4.1.

s�� 0.8, �� 0.1, y�� 1, x� 2, l� 0.5, n� 0.05.

h�� 0.1, h

�� 0.1, i

�� 1, i

�� 0.3, c

�� 0.01.

�� 0, �

�� 1,

�� 1,

�� 0.5, �� 1.1, �� 0.3.

��

�� 0.05, �

��

�� 0, t�� 0.35.

4.6.3 Investment nonlinearity and the real cycle

In this subsection we assume as in the previous one that the parametervalues �

�and �� are positive and finite and that

�� 0 holds in order to

remove monetary influences from the real part of the cycle as in section 4.4(

�� n again for reasons of simplicity).The other assumptions of section 4.4 that were used for isolating the real

cycle were:

∑ h��, i.e., r� r

�, and

∑ �� , i.e., �� n� 0.

These two assumptions here simply serve to reduce the dimension of thefull dynamics by two to two (now in the variables u��/x, l), thereby againallowing a preliminary investigation here of the real part of the model (asan isolated substructure of the full dynamics). The resulting autonomoussubdynamics in the variables u and l read

u� [(1� �)��(y(u)/(xl)� 1)� (

�� 1)�

�(y(u)/y�� 1)], (4.62)

l1 � n� s�(y(u)(1� u)� �� t�), (4.63)

with the output—capital ratio y(u) implicitly given by

s�(y(u)(1� u)� �� t�)� n� i

�(y(u)(1� u)� �� r

�)

� i�(y(u)/y�� 1). (4.64)

This model has been extensively studied in section 4.4 with respect to alinear shape of the investment function in the implicitly defined y(u) rela-tionship (and linear or nonlinear market adjustment functions). It will nowbe briefly investigated by assuming as in the preceding subsection thenonlinear shape

i�(�� r

�)� c

�tanh(i

�(�� r

�)/c

�),� � y(u)(1� u)� �, c

�� 0,

for the profitability component in investment behavior of the employedinvestment schedule i( · ) in the place of the nonlinearities we employed insection 4.4. This function of � is strictly increasing and zero at � � r

�, and it


Figure 4.16 The nonlinear component of the investment function

approaches two bounds c�,�c

�as ��. Furthermore, there exist

exactly two values �, �� of � where the slope of this function is equal to s�, if

the parameter i�is chosen larger than s

�. Figure 4.16 summarizes this

situation.�With respect to such a reformulated investment function one gets (as in

section 4.2 of this chapter) for the derivative of the function y(u) defined bythe above goods-market equilibrium condition (4.64),

y�(u)�(sc� i�

�( · ))y(u)

(s�� i�

�( · ))(1� u)� i

�/y�

�N

D. (4.65)

This derivative, and the function y(u), are locally well defined around thesteady state u

�ifD� s

�(1� u

�)� i

�/y�� 0 holds true, which is assumed in

the following.� Closer inspection of the denominator D of this derivative

See figure 4.15 for a numerical plot of this function for c�� 0.01.

� Note that we have y� y�when the above investment curve intersects the depicted savingsline.

� This assumption is the opposite of an assumption we made in the subsection on the puremonetary cycle.


Figure 4.17 A nonlinear goods-market equilibrium curve

furthermore shows that it (and the function y(u)) is well-defined on theinterval (u

�, 1), at which the value of u

�is given by the first point to the left of

u�where the denominator D vanishes (this will happen at a value u

�� 0 if

s�� i

�/y�� i�

�(y(0)� r

�) is assumed in addition). We shall work in the

following with the opposite assumption, i.e., we assume that u�� 0 holds.

Let us provisionally here also assume for the following that the functiony(u) fulfills the conditions y(1)� 0, y(0)� y�, y(u)� 0 for u� u

�. Due to

equation (4.65) it must then therefore have the general shape depicted infigure 4.17.First we need to locate the values u� , u in this figure. It is easy to obtain

from equation (4.64) (by substituting into it �� y(1� u)� ��) the resultthat the derivative of the thereby defined function �(u) is given by

��(u)�i�y/y�

(s�� i�

�( · ))(1� u)� i

�/y�.

This derivative is strictly negative on the whole interval (0, u�), u�� u�� 1

The first condition is implied by s�(�� t�)� n� i(� r

�� )� i

�.

� And y� (�� )/(1� n).


Figure 4.18 The phase diagram of a pure real cycle

where the values of the function y(u) are positive (see figure 4.17). It followsthat there exist uniquely determined values u, u� in (0, u�) such that�(u)� �,�(u� )�� , since y(u) passes once again through y� to the left and tothe right of u

�.

Let us now calculate the isocline u� � 0. As in section 4.4, it iseasily shown to be of the form l� (1/x)/((1� q)/y(u)� q/y�) whereq� (1�

�)��/(1�

�)��. It follows that this expression is always well

defined in the above situation (u � (0, u�)) when q� 1 is assumed. Thederivative of this isocline reads

l�(u)�(1� q)y�(u)

[(1� q)/y(u)� q/y�]�xy(u)�,

i.e., the slope of this isocline has the same sign as that of the y(u) curveshown in figure 4.17. It furthermore cuts the horizontal axis at the samevalue of u as the y(u) curve, though it will still be finite in value when theformer curve is infinite (at u� u

�� 0).

The assumed situation thus gives rise to the phase diagram shown in


Figure 4.19 A simulation of the pure real cycle

Table 4.2.

s�� 0.8, �� 0.1, y�� 1, x� 2, n� 0.05.

h�� 0.1, h

�� 10000, i

�� 0.9, i

�� 0.3, c

�� 0.1.

�� 4, �

�� 0.4,

�� 0.2,

�� 0.5, �� 0, �� 0.

�� 0.05,

�� 0, �

��

�� 0, t�� 0.35.

figure 4.18 for the dynamical system (4.62)—(4.63). This phase diagramsuggests that the conditions of the Poincare—Bendixson theorem are fulfil-led, since (i) no trajectory can leave the positive orthant, and (ii) a trajectorythat connectsA with A� can always be found. The set enclosed by ABCD istherefore an invariant set of the considered dynamics. Hence, due to theinstability of its unique steady-state, the limit set of each trajectory whichstarts in the domain must be a closed curve.The diagrams of figure 4.19 provide a simulation study of this partial real

limit cycle. The parameters of this simulation are shown in table 4.2. In the


Figure 4.20a A simulation of the joint monetary and the real cycle in theintrinsically nonlinear case (with no investment nonlinearity)

top right is shown the nonlinear y(u)-function that is generated by thenonlinearity in the investment function for those u-values that are in factreached through the depicted trajectory. Remarkable in these figures is alsothe fact that the rate of capacity utilization is not closely correlatedwith therate of employment of the labor force. The rate of profit, by contrast, movesstrictly inversely to the share of wages (though the function y(u) is notmonotonic).

4.6.4 The real and monetary cycles in interaction

Let us now consider briefly the above real and monetary cycles in interac-tion, i.e., the four-dimensional dynamical system (4.30)—(4.33).�By exploiting the many linear dependencies of the Jacobian of the

four-dimensional case t�� const. at the steady state as in lemma 4.1 (see

� More detailed investigations of these and other interactions of the given state variablesfollow in chapter 6 where the ‘‘working model’’ of the hierarchy of models of monetarygrowth of this book is introduced and investigated.


Table 4.3. Parameter set for figure 4.20a

s�� 0.8, �� 0.1, y�� 1, x� 2, n� 0.05.

h�� 0.1, h

�� 0.1, i

�� 0.9, i

�� 0.3.

�� 4, �

�� 4,

�� 0.2,

�� 0.5, �� 1.1, �� 0.3.

��

�� 0.05, �

��

�� 0, t�� 0.35.

also chapter 6 for further calculations of this type in the case of a dynamicalsystem of an even higher dimension), it is not difficult to show that thedeterminant of this Jacobian is always positive. Stability can therefore onlybecome lost in a cyclical fashion, by way of a Hopf bifurcation in general.��Furthermore, situations where the steady state of the dynamics is indeedlocally asymptotically stable can be obtained by extending the proposi-tions of the three-dimensional case of the preceding section to the presentsituation with medium-run inflationary expectations by assuming par-ameter values for �� that are sufficiently small and by applying continuityarguments with respect to the real parts of the eigenvalues of the corre-sponding three-dimensional subcases.When stability gets lost through an increasing parameter ��, and there

is for example not an attracting limit cycle that keeps the dynamicsbounded in such a case, the question arises as to what are the furthereconomic nonlinearities that can be meaningfully employed to make theexplosive dynamics viable. Such further constrainingmechanisms can ariseas thresholds, as PID (proportional, integral, derivative) feedback control-lers, through increasing flexibility of appropriate variables as the systemmoves further and further away from the steady state, etc. We will herechoose the first type of nonlinearity, in the employed investment function,for a first simulation of its effects on the four-dimensional dynamics of thischapter.We have seen that nonlinearities in the investment function are difficult

to treat analytically even in the two partial and autonomous two-dimen-sional cases we have considered above. The present stage of the investiga-tion of the interaction of these two-dimensional dynamics thus allows onlyfor numerical presentations of this full dynamical system in the presence ofthe investment nonlinearity. Figures 4.20a and 4.20b provide, on this basis,a first brief impression of how an explosive situation in the linear case maybe tamed, at least for some time, by the only nonlinearity (i.e. in theinvestment function) we have considered in this section.The parameter values for figure 4.20a with only natural nonlinearities

�� The role of price and inflationary expectations flexibilities is here again of decisiveimportance.


Figure 4.20b A simulation of the joint monetary and the real cycle in theextrinsically nonlinear case

Table 4.4. Parameter set for figure 4.20b

s�� 0.8, �� 0.1, y�� 1, x� 2, n� 0.05.

h�� 0.1, h

�� 0.1, i

�� 0.9, i

�� 0.3, c

�� 0.1.

�� 4, �

�� 4,

�� 0.2,

�� 0.5, �� 1.1, �� 0.3.

��

�� 0.05, �

��

�� 0, t�� 0.35.

are given in table 4.3. We see from figure 4.20a that the dynamics areexplosive over a fairly short time horizon.In the simulation of figure 4.20b we introduce the nonlinear investment

function by using the tanh-function of equation (4.60) with the value 0.1 forthe parameter c

�, and keep all other parameter values unchanged. For easy

reference the full parameter set is displayed in table 4.4. The plots in figures4.20a and 4.20b show at the top the real and the monetary ‘‘cycle’’ we havestudied in the preceding subsections in isolation from each other.


The plots of figure 4.20b show that the explosive cycle in figure 4.20a canindeed be ‘‘tamed’’ to some extent and for some time. It is however alsoapparent that further extrinsic nonlinearities are needed here in order toget a dynamical system with truly bounded trajectories. We shall not gointo this question here any further, but shall proceed in the followingchapters to further extensions of the Keynesian prototype dynamics estab-lished in this chapter where the question of the ‘‘viability’’ of the employeddynamics will be posed anew and considered from a more advanced andmore refined perspective.Before closing, we note finally that the plot in the lower right hand box of

figure 4.20b shows the time series of two pairs of variables that are oftenidentified and displayed solely by one magnitude in macroeconomic rea-sonings and presentations, namely, on the one hand, the rate of employ-ment of the labor force V vs. the rate of capacity utilizationU of firms and,on the other hand, the rate of price inflation p vs. the rate of wage inflationw. We consider it important that these variables are introduced and theirdynamical laws investigated independently of each other (as to the degreeof independence they can really have from each other). We also believe it isimportant that conditions are explicitly provided (and justified with re-spect to their empirical contents) which allow the use, on the one hand, ofonly one variable to represent ‘‘capacity utilization’’ and, on the otherhand, of only one variable to represent ‘‘inflation.’’ Such an analysis wouldprovide reasons for (or against) the validity of Okun’s Law, on the onehand, and, on the other, for the extent to which the reliance on simplemarkup pricing rules is really justified in models of the interaction ofunemployment and inflation.

4.7 Outlook: adding smooth factor substitution

When we motivated the basic modification of the Keynes—Wicksell modelthat led us to the present chapter we stated that proper Keynesian modelsof monetary growth have rarely been discussed in the literature on monet-ary growth. There is, however, a seemingly important exception to thisrule, given by the Keynesian AS—ADmodel of monetary growth as it is forexample discussed in detail in Sargent (1987, ch. 5). This model typeemploys IS—LM-equilibrium, and the so-called AD curve that derives fromit, as well as the AS schedule representing the situation where prices equalmarginal wage costs. For medium-run analysis it then adds a money wagePhillips curve and a scheme for expectations formation and for long-runanalysis of the conditions for factor growth.Yet, if one considers this model thoroughly, one can again find that this

model type is not really of a Keynesian nature, since the capital stock is


fully employed as in models of Keynes—Wicksell type. This is due to the factthat firms always operate at their profit maximum in this AS—AD growthmodel, and thus are on their supply schedule at each point in time. But,again, a Keynesian theory of fluctuating growth should allow for an under-or overutilized capital stock besides the under- or overemployment oflabor, and thus for capital and labor to be off their supply schedules. Sincethe AD—AS growth model employs a neoclassical production function andallows on this basis for smooth factor substitution, the task now is inparticular to formulate this proper Keynesian approach to fluctuatinggrowth in the presence of a technology which allows smooth factor substi-tution in place of the fixed proportions we have assumed so far.It is the purpose of the following chapter to do exactly this and to show

on this basis that the AS—AD growth case represents but a bastard limitcase between the general Keynes—Wicksell and the proper Keynesianmodel of monetary growth, where both capital and labor are generallyover- or underemployed, as the economy evolves. Furthermore, and forcompleteness, we shall also investigate the neoclassical approach (theTobin model type) in the next chapter for the case of smooth factorsubstitution which, as the Keynes—Wicksell variant, was and remains amodel with full capacity growth. Yet, at the core of this chapter is thedevelopment of a prototype model with a varying degree of labor as well ascapital utilization in the presence of smooth factor substitution. To do this,the AS schedule is reinterpreted as the locus of potential output with whichthe actual output of firms subject to a Keynesian effective demand regimehas to be compared in order to define the rate of capacity utilization on thebasis of a given neoclassical production function.This is the proper interpretation of the AS schedule which makes this

schedule a reference schedule with respect to which there is price adjust-ment in the medium run along the same lines as in the present chapter onKeynesian monetary growth.Models of monetary growth of Tobin or Keynes—Wicksell type have

generally been considered on the basis of smooth factor substitution. Thenext chapter, therefore, only shows in this regard how this assumption is tobe formulated in our general prototypemodels of this type, leading to moreflexibility, further adjustment processes, and more stability in these setupsbasically. With respect to the Keynesian approach to monetary growth ofthis chapter, the next chapter will, however, show that this type of addi-tional flexibility does not add much to the proper understanding of Key-nesian IS—LM growth analysis. This is in contrast to the widely held beliefthat smooth factor substitution will undermine basic assertions of a Key-nesian analysis of growth, stability, and business fluctuations. The nextchapter will therefore demonstrate that the perspective of chapter 4 is not


changed in an essential way through the introduction of smooth factorsubstitution.

Appendix 1: The Benassy business cycle model

It is useful to compare the limit cycle results of section 4.4 with another wellknown cycle model of the IS—LM variety, namely Benassy’s (1986a) non-Walrasian model of the business cycle. On the one hand, we can useBenassy’s proof of the conditions of the Poincare—Bendixson limit cycletheorem to fill out the details in our applications of this theorem. On theother hand, contrasting the limit cycle generating mechanism of the Be-nassy model with the mechanisms we have used in the previous sectionmay help to clarify the advantages and disadvantages of these differentderivations of business cycle fluctuations.The Benassy (1986a) IS—LM cycle model�� assumes no growth, i.e.,

L�L� ,K�K� (l1 � 0), and smooth factor substitution which is governed bythe marginal wage cost pricing rule in the usual way, i.e., p�w/F

�(L,K� ).

As we shall see in more detail in the next chapter (section 5.4), this resultassumes that prices are adjusting with infinite speed if there is any devi-ation from full capacity utilization, so that such capacity mismatch isalways avoided thereby.�� For a Keynesian model it is, however, moreappropriate to allow for capacity utilization problems and to check whatadditional features of the model may be generated thereby. We thus makeuse again of the simplifying assumption of fixed proportions in productionin establishing also thismodel’s basic structure. This allows for the immedi-ate inclusion of capacity utilization problems (as we know from the rest ofthis chapter), which have not been considered in Benassy’s (1986a) paperapart from his use of the acceleration principle.In order to stay close to Benassy’s two-dimensional approach to econ-

omic dynamics and his use of the nominal wage rate as one of the dynamicvariables we add the usual markup pricing formula (in the place of theabove marginal productivity rule) to the present formulation of the model,which in particular implies that there is no difference between wage andprice inflation as is usually assumed in Phillips-curve approaches to thedetermination of the rate of inflation. In sum, we therefore have

L�Y/x,Y� �� y�K� , p� (1� z)wLY

� (1� z)w

x,x, z� const.

�� Compare section 5.4 for a treatment of the smooth factor substitution case.�� The basic problem here however is why there is an accelerator mechanism as in Benassy

(1986a) in the presence of perpetual full capacity utilization.


for the description of production and pricing in place of Benassy’s originalassumptions.Benassy (1986a) uses the following effective demand block for the deter-

mination of equilibrium output Y and the rate of interest r:

C(Y, p)� I(Y, r) ��Y, (4.66)

pm(Y, r) ��M� , (4.67)

where Y is the demand expected by investors when planning their currentinvestment decision. Since growth is disregarded in the Benassy model(Y�� const.), the variable Y is here reinterpreted and reformulated as theexpected degree of capacity utilization U�Y/Y� �, which takes the placeof the actual degree of capacity utilization we have used in the Keynesianmodel of this chapter.�As a further necessary ingredientwe need a specification of the formation

of demand and capacity utilization expectations of investors. This is donein the form of the following standard adaptive expectations mechanism

U� � �)�(U�U),U�Y/Y� �.

Finally, there is a money wage Phillips curve in the Benassy model of thesimple form

w� �(L/L� ),�� 0,�(0)� 0,�(1)��,

which here translates immediately into the usual price-inflation form bymeans of the above assumption of a markup pricing of firms p(� w)��(Y/(xL� )). This means that either the price level or the level of nominalwages can be used in the final formulation of the dynamics of the model.The IS—LM equations imply for output and the interest rate the func-

tions Y(U,p), r(U, p), by means of the usual assumptions on such a de-mand block; see Benassy (1986a, pp.134ff.). This in turn implies that wehave to deal with the autonomous two-dimensional dynamical system

U� � �)�(Y(U, p)/Y� ��U),Y

)�� 0,Y

�� 0, (4.68)

p��(Y(U, p)/(xL� )),�� 0,�(V�)� 0, (4.69)

and to explore its local and global stability properties. The steady state isdetermined by Y

��V

�· (xL� ) and Y(Y

�, p

�)�Y

�(see Benassy 1986a,

pp.138 ff.) with respect to its existence and uniqueness. For the Jacobian atthe steady state we have

� The influence of the rate of profit � and of the (expected) rate of inflation � as they arepresent in our Keynesian model is neglected in Benassy’s formulation of investmentbehavior. The (destabilizing) Mundell effect (Y�� 0) in particular is thus excluded there.Note that Benassy instead includes a (stabilizing) Pigou effect (C

�� 0) into his formulation

of IS—LM equilibrium.


Figure 4.21 Benassy’s money wage Phillips curve

J��)�(Y)�/Y� �� 1) �

)�Y�

��Y)�/(xL� ) ��Y

�/(xL� )� . (4.70)

Benassy assumes that at the steady state the condition� �)�(Y)�/Y� ��

1)��Y�/(xL� )� 0 holds (which follows if Y

)�/Y� �� 1 and �

)�is chosen

sufficiently large), and thus guarantees its local instability. For the shape ofthe Phillips curve he moreover assumes (in order to obtain global stabilityfor the considered dynamics) the general shape shown in figure 4.21.Figure 4.22 shows the phase diagram that Benassy then derives from

such a situation. Note that in this figure the labels 1, 1� �, and V�denote

the curves which satisfy Y(U, p)� const. · (xL� ), and where const. is setequal to 1, 1� �, and V

�respectively.

Benassy shows that the vector field corresponding to (4.68)—(4.69) mustpoint inward along the (1� �) curve if � is chosen sufficiently small (sincep�� along the 1 curve), and thus gets that the (1� �)&(1� �)&B&Ddomain is an invariant subset of the dynamics (if the point D is chosen suchthat it lies below the p� � 0 isocline). The properties of this domain thenimply that there exists at least one limit cycle in this domain around thesteady state of the model.��

� This amounts to the assumption of a Harrodian cumulative instability situation in thesetup of a full capacity utilization model as far as the original Benassy model is concerned.

�� Benassy’smethodof proof is also applicable to theRose-type situationswe have consideredabove and can be usefully employed in the formulation of the proofs of such limit cycleresults. We have only sketched such proofs in section 4.4 and in this appendix.


Figure 4.22 Construction a viability domain for the Benassy model

The outcome of the Benassy cycle model thus is to turn the explosivedynamics — caused by demand expectations of investors that are sufficient-ly strong in their impact on investment behavior (and thus on effectivedemand) — into a viable one if there is nearly infinite upward nominal wageflexibility near the full employment ceiling. The Keynes and the Pigoueffects then induce contractionary forces near this full employment ceilingthat successfully counteract the destabilizing influence of the demandexpectations of firms.The locally destabilizing role of the Mundell effect Y� is obtained in the

Benassy model by the accelerator term U in the investment function,which must have an impact effect on goods-market equilibrium which islarger than 1 in order to obtain the local instability result. As alreadystated, global stability is then achieved by the assumption of a one-sidednonlinearity in the labor market, since an increase in wages is alwayscontractionary in the present context and thus does not stimulate effectivedemand as was possible in section 4.4.What then finally distinguishes this limit cycle result from the corre-

sponding results in section 4.4 and earlier sections of this book? We firstnote here that there is no real-wage mechanism (neither a Goodwin type


one nor the Rose type one) necessary for the mechanics of the Benassymodel. Instead, real wages may be safely assumed to be constant in time (aswe have indicated above) without sacrificing an essential feature of theBenassy cycle model. The cycle of this model is instead generated by anaccelerator investment mechanism (which must operate with sufficientstrength), combined with the globally stabilizing role of the Keynes (andthe Pigou) effect due to rapid wage increases when the full employmentceiling is approached.��Benassy’s model thus is basically a so-called AD—AS model (where the

A(ggregate) S(upply) function may be a horizontal curve) augmented by alocally destabilizing accelerator mechanism. The basic message of themodel therefore is that, whatever the reasons for local instability, theKeynes effect (cum Pigou effect if desired) is a sufficiently strong mechan-ism to keep the model a viable one provided there are sufficiently strongprice effects near the full employment ceiling. We stress here that one suchnonlinearity is already sufficient in this context to create bounded dynami-cal behavior. This appendix thus provides a bridge to the monetary forcesadded in the later sections of this chapter to the real growth dynamics onwhich we focused solely in the earlier sections. The resulting system (in theabsence of Mundell effects on aggregate demand and inflation accelerationterms in the Phillips curve mechanism) also exhibits considerable potentialfor stabilizing the economy, as we saw in section 4.5 for the monetarygrowth dynamics model of dimension three.

Appendix 2: Technical change, wage taxation, averageinflation, and p-star expectations

In this appendix we sketch how the model of section 4.1 can be extended invarious directions that make it more plausible from the empirical point ofview while remaining in the framework of its four laws of motion for theprivate sector of the economy.The first extension is to a fixed proportions technology which exhibits

Harrod-neutral technical progress at a constant rate n�. This takes account

of the fact that labor productivity x as well as labor intensity l exhibit,from a long-run perspective, a positive trend in capitalist market econo-mies. They thus cannot be treated as given constants when the aim of theinvestigation is to reproduce some of the stylized facts of growth patternsand business fluctuations. Harrod-neutral technical progress is in the caseof a fixed proportions technology defined by a given potential output—

�� There is no further nonlinearity necessary here for the achievement of global stability.


capital ratio y��Y�/K and a growing output—employment ratio x�Y/L, with a given growth rate x� n

�which, in addition, is here assumed to

be independent of the level of output that is actually produced. Theseassumptions imply that the actual ratios k�K/L, l�L/K (describingactual capital and labor intensity) will change over time, not only becauseof fluctuations in the utilization rate of the capital stock, but also in thesteady state where we will have k1 � n

�� l1 .� These assumptions and

their implications are part of Kaldor’s list of the stylized facts of growththeory (see Jones 1975, pp.7—8, for example).The prototype Keynesian model is further extended by (i) allowing now

for a constant wage income tax rate, (ii) letting medium-run inflationaryexpectations now be determined by an average of price and wage inflation(instead of only price inflation), and (iii) by basing the forward-lookingcomponent on a basic version of the so-called p-star concept of the Ger-man Bundesbank (and other central banks) in order to allow for a morerefined treatment of forward-looking expectations. All these extensionswillextend the final dynamical systems they imply only in a straightforwardand clearly understandable way, making the dynamics richer but not lesstransparent.These new assumptions on the technology of our economy imply the

following changes in the model of section 4.1. Labor productivity x nowfollows a growth trend with constant rate n

�, i.e., the variable x is now

determined by x� x�exp(n

�t). Trend growth in the output of the economy,

with the rate of employment a fixed magnitude, will therefore now occur atthe rate n� n

�, i.e., the growth in labor supply plus that in labor productiv-

ity. This new ‘‘natural’’ growth trend of the economy consequently has tobe used in the money supply rule, in the trend component of the investmentschedule, and in the formulation of inflationary expectations replacing theformer natural growth rate n.�� Finally, the wage—price module of oureconomy has also to reflect the growth in labor productivity, on the onehand, through an augmented target of workers that, besides expectedcurrent and medium-run inflation, now is also based on the constantincrease in labor productivity that they observe. On the other hand, thepartial markup pricing objective assumed to underlie the term

�w in our

price formation rule now gives rise to the expression w� n�as the cost-

push reference for firms, which thus has to be used in the place of w in thisequation for the formation of the rate of price inflation.

� Note in what follows that l and l will be used to denote labor intensity magnitudes inefficiency units.

�� Note that this new trend growth rate is also the most natural one to be used for theparameters

�and

�, which again implies that the steady state is then inflation free and

debt free.


The most significant change in the dynamics takes place in theequation which describes the formation of inflationary expectations:�� (p� �)� ��(�

� (n�� n

�)� �). On the one hand, we replace

the adaptive term in this adjustment equation by ��(�p� (1� �)(w�

n�)��)��(�(p��)� (1� �)(w� n

��)), which states that adaptive

expectations are now based on an average of price and wage inflation (thelatter only insofar as this inflation rate deviates from the trend in laborproductivity growth). Note here that the earlier situation is reproducedwhen � � 1 is assumed. The rationale for this extension is that economicagents believe to some extent that wage inflation will become price infla-tion in the end (if it exceeds the rate of productivity growth), and that theytherefore integrate aspects of wage inflation into their views on medium-run inflation.��On the other hand, we assume with respect to the forward-looking

component in the formation of inflationary expectations that economicagents adopt the so-called p-star concept used by some monetary authori-ties in their calculation of future price changes.�� A basic version of thisconcept is as follows: Economic agents assume that the price level isdetermined in the longer run by the quantity theory of money and bynormal output, i.e., the moving attracting equilibrium of future price levelsis assumed at each moment in time to be given by p*� v�M/(U� Y�), where v�is the velocity of money (estimated or assumed as a constant). Since there isno inside money in the present model, the magnitudeM is fully under thecontrol of the central bank. The only remaining problem for the bank aswell as the other economic agents then is to estimate the development ofnormal output, which in our model is given by potential output times thedesired rate of capacity utilization.In terms of growth rates the above equation gives rise to p*�

��K1 ,

due to our assumption of a fixed proportions technology. The monetaryauthority as well as the public therefore has in this case only to obtainknowledge of the growth rate of the capital stock in order to calculate thispresent rate of growth of the center of gravity of the price level, rightly orwrongly assumed to exist for the actual rate of inflation. We here assumethat the actual rate of growth of the capital stock is common knowledgeand thus end up with the following reformulation of the forward-looking

�� It is also not difficult to formulate a separate equation for expected price andwage inflation.However, the dimension of the dynamics is thereby increased by one.

�� We owe the knowledge of this concept to D. Malliaropulos, who, however, is notresponsible for the particular form given to it in the following. The quantity theoreticapproach here employed can be obtained from our formulation of the money market bysetting the parameter h

�equal to zero, which gives as money-market equilibrium the

equation M� h�pY� h

�pY�U. Replacing U by its steady-state value U� then gives the

equation shown for the estimation of p*.


term in our standard equation for the formation of inflationary expecta-tions: ��(p*��)� ��(�

�K1 ��).Note that this term reduces to our earlier formulation of forward looking

inflationary expectations when it is assumed that both the money supplyand the capital stock grow at their steady rates. Note also that this is only asecond step toward a more advanced theory of future inflation. Furthersteps necessarily must use more elaborate models of price forecasting thanhave been introduced here, which take more of the present structure of themodel into account than only the long run quantity theory of money towhich it gives rise to (v� � 1/h

�!). This, however, is only a matter of sophisti-

cation (which from a numerical point of view will create no real additionaldifficulties), and not a matter of a completely different approach to theformulation of the theory of inflationary expectations.A final extension contained in the following model is that it is now

assumed that wage income is taxed proportionally. This extension does notchange the dynamical equations of the model and mainly serves to give acloser correspondence of the model structure with actual taxation pro-cedures.��

The equations of theKeynesian IS—LMgrowthmodel with the above set ofextensions are:


��w/p, u��/x,� � (Y� �K��L)/K, (4.71)


%� 1. (4.72)



�pY� h

�pK(1� �

�)(r

�� r),

(4.73)

C� �L�T�

� (1� s�)[�K� rB/p�T

�], (4.74)

T�T��T

�,T

�� (1� �

�)�L, (4.75)

S��Y� �K� rB/p�T�C� s

�[�K� rB/p�T

�]� s

�Y��

� (M� �B� � pE� )/p, (4.76)

L1 � n�� const. (4.77)


Y�� y�K, y�� const.,U�Y/Y�� y/y�(y�Y/K), (4.78)

�� See Flaschel, Gong, and Semmler (1998) for a much more advanced treatment of the taxand transfer policies of the government.


L�Y/x, x� n�(x� x

�exp(n

�t)),V�L/L�Y/(xL), (4.79)

I� i�(�� (r� �))K� i

�(U�U� )K� K, � n

�� n

�, (4.80)

pE� /p� I, (4.81)

K1 � I/K. (4.82)


T� ��L�T

�, (4.83)

T��

�(�K� rB/p), (4.84)

G�T� rB/p��M/p, (4.85)


M1 ��, (4.87)

B� � pG� rB� pT�M� ][� (��

�)M]. (4.88)



�pK(1� �

�)(r

�� r), [B�B,E�E], (4.89)

pE� (1� �

�)�pK/((1� �

�)r��), (4.90)

M� �M� ,B� �B� [E� �E� ]. (4.91)


S� pE� � S

��S

��Y� �K�C�G� I� p

E� . (4.92)


w� ��(V�V� )�

�(p� n

�)� (1�

�)(�� n

�), (4.93)

p��(U�U� )�

�(w� n

�)� (1�

�)�, (4.94)

�� (�(p��)� (1� �)(w� n��))��(�

� I/K��).(4.95)

This Keynesian model with Harrod-neutral technical change is formallyidentical to the Keynesian model without technical progress as far as thismodification of it is concerned. This can be seen as follows. Equations(4.93) and (4.94) can be rearranged in the usual way, now giving rise to

w� �� n��

�(V�V� )�

�(p��),

p�� (U�U� )�

�(w�� n

�).

Solving for the two variables w� �� n�, p��, these two equations imply

w� �� n�� [�

�(V�V� )�

��(U�U� )],


p�� [��(U�U� )�

��(V�V� )],

where both of these equations will now be used in the formulation of thedynamic laws for �. Subtracting the second from the first equation further-more implies for the share of wages u�wL/(pY):

u� w� p� n�� [(1�

�)��(V�V� )� (

�� 1)�

�(U�U� )],

which gives the first of the differential equations employed in the followingin the same form as in the earlier models without technical change (where�� u held).We denote by L,L labor supply and employment measured in effi-

ciency units, i.e., L�L exp(n�t) and by, L�Lexp(n

�t) denote effective

labor supply and effective employment as distinct from labor supply andemployment in ‘‘natural’’ units. From calculations as in the precedingchapters we then obtain from the model (4.71)—(4.95), the five-dimensionaldynamical system, now expressed in terms of the variables u��/x, l�L/K, m�M/(pK), � and b�B/(pK),��

u� [(1� �)��X�� (

�� 1)�

�X�], (4.96)

l1 � n�� n

�� s( · )�� i

�( · )� i

�( · ), (4.97)

m� �� (n

�� n

�)� [�

�X��

��X�]� l1 , (4.98)

�� (�(p��)� (1� �)(w� n��))

� ��(�� (n

�� n

�)� �� l1 ), (4.99)

b� � (��

�)m� (�� n

�� n

�)b� b( (�

�X��

��X�)� l1 ),

(4.100)

where, besides the expressions derived above, we have to employ theabbreviations

�� y(1� u)� �, u��/x,l�L/K� exp(n

�t)L/K� exp(n

�t)y/x� y/x

�(y not const.!),

X��V�V� ,V� l/l,X��U�U� ,U� y/y�,r� r

�� (h

�y�m)/(h

�(1� �

�)),

t�T/K� ��l� �

�(�� rb)� t

�� t

�,

g� t�� t

�� rb�

�m� t��

�m,

t��

�(�� rb)� rb� t

�� rb, t�� t

�� t�

�,

s( · )�K1 � s�(1� �

�)(�� rb)�

�m� i( · )

i( · )� i�(�� r��)� i

�(U�U� )� n

�� n

�,

�� Note thatwemake use of the letter l to denoteL/K, both for simplicity and in order to keepthe notation close to that of the preceeding sections.


and where again actual output per capital y�Y/K(� y��Y�/K) is ob-tained by solving the IS—LM relationship

s( · )� s�(1� �

�)(y(1� u)� �� rb)�

�m� i( · )

� i�(y(1� u)� �� (r

�� (h

�y�m)/(h

�(1� �

�)))��)

� i�(y/y��U� )� n

�� n

�nowwith u��/x, the share of wages in gross national income in the placeof �, the real wage. We note that the steady state of this dynamical systemis the same as in the five-dimensional Keynesian monetary growth modelof section 4.1.2 if n

�� n

�is set equal to n (and x� x

�), since the tax rate for

wage income does not influence the interior point of rest of the dynamicsand since all remaining terms reduce either to their former expressions orto zero in the steady state.In view of these equations it is obvious that this dynamical system is of

nearly the same type as the corresponding Keynesian model withouttechnical change, wage taxation, and the other changes we have made tothe model, the major innovation being the inclusion of l1 in the forward-looking component of inflationary expectations. The model is thus, on theone hand, closer to reality than the one of section 4.1, but nevertheless, dueto its construction, has a structure of basically the same intensive form.This is in particular true with respect to the inclusion of wage taxationwhich, on the one hand, is necessary due to the empirical reasoning, butwhich, on the other hand, here only means a redistribution of incomebetween the samemarginal (� average) propensity to consume of workersand the government. Concerning technical change, the use of Harrodneutral technical progress, as is well known, needs nothing more than areinterpretation of the magnitudes we have employed so far in this chapter,in order to include the stylized fact of positive trend in labor productivityinto our models of monetary growth. We do not go into a theoreticalinvestigation of this extendedmodel here, but only claim that its analyticalproperties do not depart very much from those of the Keynesian prototypemodel investigated in the preceeding sections of the present chapter, as wehave indicated above.


5 Smooth factor substitution: asecondary and confused issue

In this chapter we recapitulate the prototype models of chapters 2, 3, and 4by adding to them smooth factor substitution in the place of fixed propor-tions in production. Our main findings will be that

• the Tobin prototype models (discussed in section 5.1) will generally� beincreased in their dynamic dimension by one, now exhibiting the dy-namics of the (full employment) labor intensity in addition;

• the Keynes—Wicksell prototype model (discussed in section 5.2) willexhibit further stabilizing mechanisms by the inclusion of a neoclassicalproduction function;

• the Keynesian model (discussed in section 5.3) will not be changed in itsgeneral qualitative features, still exhibiting underutilized labor as well ascapital, even with the addition of a neoclassical production function.

We thus find that smooth factor substitution does not essentially modifythe distinctions we have drawn between the three prototype models con-sidered in the preceding chapters. It is worth highlighting this result in viewof the fact that the fixed coefficient assumption is often criticized in theliterature for the narrowness of results it seems to imply. It is certainly true,however, that the addition of smooth factor substitution makes the modelssomewhat more complicated to handle due to the extra flexibility thissubstitution principle adds to them. The advantage of the assumption offixed coefficients in production therefore is to make the fundamentaleconomic principles of each of these theories of monetary growth moretransparent and more easily understandable. We shall hence retain thisassumption in the chapters that follow this digression on smooth factorsubstitution.This generalization of the production technology of the considered

economy has the additional effect of making the models used in chapters 2

� Up to the cases where the labor market is in disequilibrium.

242

and 3 easier to compare with their forerunners in the literature. Finally, wecan demonstrate through this extension in section 5.3 that the standardtextbook model of Keynesian dynamics (see Sargent 1987, ch. 5, forexample), is but a limit case of both the Keynes—Wicksell supply-sideapproach as well as of ourKeynesianmodel where, however, the features ofthe Keynes—Wicksell model will dominate. This bastard situation willagain demonstrate that the usage of Keynesian models in the orthodoxliterature is strongly biased toward supply side economics, and thus notwell suited to understandproperly even the basics of Keynesian economics.Including smooth factor substitution is thus not only a secondary issue

with respect to the working of the models, but has also often led toconsiderable confusion in the presentation, interpretation, and investiga-tion of Keynesian monetary growth dynamics.

5.1 The Tobin case: one further integrated law of motion

5.1.1 The general equilibrium case

5.1.1.1 Adaptive expectationsLet us first reconsider the basic general equilibrium subcase of the Tobinmodels of chapter 2 which we have investigated there in section 2.1 for thecase of fixed proportions in production by means of one dynamical law.The only changes in the equations of this model concern equations (2.8)and (2.15). Under the assumption of smooth factor substitution (2.8) is to bereplaced by Y(�Y�)�F(K,L), where the function F has the usual prop-erties of a neoclassical production function (see, e.g. Sargent 1987, pp.7ff.).Equation (2.15) is then justified by the marginal productivity ruleF�� w/pwhich is the motivation for the assumption that the equality

L�L( /K1 � n!), i.e., full employment holds at each point in time (the‘‘const.’’ in (2.15) is generally set equal to 1 in such a context!).By virtue of Euler’s theorem on homogeneous functions, we then get

from (2.1) for the rate of profit � the equality ��K�. In intensive form

(Y/K� y� f (l), l�L/K, see Sargent 1987, p.118, for details) we therebyget for the discussion of the dynamics of the revised model the newrelationships

y� f (l),� � f �(l), �� f (l)� f �(l)l��(l),

where f �(l)� 0, f �(l)� 0 and ��(l)� 0 hold true.Since the ratio l�L/K�L/K is no longer fixed (K1 �L1 � n in gen-

eral), its movement in time must be explained, and this is done by means ofthe dynamic law

l1 �L1 �K1 � n� (y� � � c� g� ) (using 2.14)

243Smooth factor substitution

� n� s�(�(l)� g� )� (1� s

�)(

��)m, (5.1)

since y� f (l), c��l� (1� s�)[�(l)�m�� t], (from (2.4) and (2.5)) and

(from (2.11) and (2.9)) t� g� � �m (� and g� are given magnitudes).

Furthermore, equation (2.12) for money-market equilibrium now readsm� h

�f (l)� h

�(r��(l)� �), from which

�� r��(l)�

h�f (l)�m

h�

��(l,m). (5.2)

Finally, the expectations revision mechanism (2.16) of section 2.2 gave risein that context to the dynamic law (see equation (2.23))

m�1

��(m)m��[(�� )(�

� n��(m)].

This equation is no longer appropriate, since � now depends onm and l andsince K1 � n (which in turn implies l1 � 0) no longer holds. In the presentsituation, this expectations formation mechanism instead gives rise to �� l� � �

�m� ��(�

� n� l1 � m��)� ��(�� n� �), since we have by

the definition of m�M/(pK) that p�� n� l1 � m must be true. This

last equation can be rearranged to yield

m��

�l� ��

��m��

l1 �� m��

(�� n� �). (5.3)

Taken together, equations (5.1)—(5.3) constitute an autonomous system ofdifferential equations of dimension two (in the variables l and m). Thissystem replaces the single dynamic equation (2.22) (or its equivalent interms of m namely equation (2.23)) of the model of section 2.1. It takesaccount of the fact that labor intensity l�L/K is now a variable of themodel (the movement of which is explained by two independent laws ofmotion for the variables L andK), and that the substituted variable � nowalso depends on this labor intensity l (via y and �). This dynamic system hasnow to be investigatedwith respect to the implications to which it gives risebecause of the added flexibility in y, l, and �.Note here, first of all, that equation (5.3) is a direct generalization of

equation (2.23) we have derived in the nonsubstitution case (l1 �0,� ��(m)). Note, furthermore, that the partial derivative m

�is given by (at

the steady state)

m��

�� l

�� m

�

l1��

��

�m

�

· ��,


where l1�� (1� s

�)n� (1� s

�)��m

�� 0, (�

�� 1/h

�� 0). The sign of �

�is, however, ambiguous, since �

��(l)� h

�f �(l)/h

�(it is negative if h

�is

chosen sufficiently small). Independently of the sign of this partial deriva-tive we can here, however, state that m

�must become positive if the

parameter �� is chosen sufficiently large (noting that ��m

�� 0).

In addition, we have for l1�the expression l1

�� s

��(l

�)� (1� s

�)��m

�,

which again is ambiguous in sign. This expression, however, does notdepend on �� which implies that the trace of the Jacobian J of the givendynamical system can always be made positive by choosing �� sufficientlylarge. This reproduces the instability result of section 2.2 (for adaptiveexpectations) as far as the trace of the present two-dimensional dynamics isconcerned.Yet, in section 2.2, we had trace J� det J� 0 (in dimension one), while it

is a common observation that the two-dimensional monetary growthdynamicsmodel exhibits saddlepath stability, i.e., that det J� 0 holds. Thequestion therefore remains whether this condition can be established forthe dynamical system (5.1)—(5.3) as well.To show this, we note first that

det J� �l1�

l1�

�a��

�a�� , a� (�� )/(��

��)

must hold at the steady state. Some simplifications yield that

det J� ��s

�� (1� s

�)n

�a��

�a�� s

�a��

�� a(1� s

�)n�

�.

Nagatani (1970, p.172) assumes h�f �(l)� h

��(l)� 0 for the (more general)

money demand function he employs. This condition is equivalent to �� 0

which (together with �� 0) then implies det J� 0 (if a is made positive

through an appropriate choice of ��). Note again, that the Nagatanicondition �

�� 0 can always be fulfilled by choosing the parameter h

�sufficiently small.Our findings in this section on adaptive expectations thus are that the

results for the case of fixed proportions in production are confirmed andfurther qualified in the smooth factor substitution case, now also allowingfor an adjustment of labor intensity l when the economy is outside thesteady state. Nothing essential is, however, added to the earlier model bythe integration of flexibility in the choice of factor proportions. Assumingsuch proportions as fixed has the advantage of making the model and itsdynamics much more transparent, and thus helps to clarify the basicmechanisms that are at work in such models of monetary growth.


5.1.1.2 Perfect foresightIn the case of myopic perfect foresight (p��) we obtain from equations(5.1) and (5.2) of the preceding subsection the two dynamical laws

l1 � n� s�(�(l)� g� )� (1� s

�)(

�� p)m, (5.4)

p� r�� (l)� (h

�f (l)�m)/h

�, (5.5)

where we again have by definition of m the further dynamic relationship

m� �� n� l1 � p (or p�

�� n� l1 � m).

Inserting this latter relationship into (5.4) and (5.5), and solving for l1 and m,then gives rise to the dynamical system

�1�(1�s

�)m �(1�s

�)m

�1 1 ��l1

m��n�s

�(��g� )� (1�s

�)nm

��n�r

��(h

�f (l)�m)/h

��or (noting that det [. . .]� 1)

�l1

m�� 1 (1� s

�)m

1 (1� s�)m� 1� ·�

a�(l,m)

a�(l,m)�

where a�and a

�denote the right hand side of the preceding equation

system.Noting that a

�and a

�are zero at the steady state, we readily calculate the

determinant of the Jacobian J of this system at the steady state to be

det J� ��s

��l

�(1� s

�)nl

�m

�(�� h

�f �/h

�) m

�/h

� � ,which is identical to the determinant of the preceding subsection up to theexpression a� (��)/(��

m��), which is equal to 1 for �� .

The Nagatani condition therefore once again implies det J� 0, and thus asaddlepath situation at the steady state of the above dynamics. Note herethat this condition is stronger than is strictly necessary in order to obtainsaddlepath dynamics.The general conclusion of this subsection is the same as in the preceding

one. Assuming smooth factor substitution extends the dimension of thedynamics to be considered by ‘‘1,’’ but does not add new features to it. Thebasic mechanism that is at work here is therefore best understood whenfixed proportions in production are assumed. These conclusions extend tothe general equilibrium Tobin model with bonds, which, however, is nottreated here anew.As in the consideration of the literature on the Tobin model in chapter 1,

all calculations can, of course, also be done by making use of k�K/L


instead of l�L/K as the basis of the intensive form formulations. In thiscase, one has to make use of x� f (k),x�Y/L,� � f �(k),�� f (k)� f �(k)kin the place of y� f (l), y�Y/K,� � f �(l),� � f (l)� f �(l)l in the equationswe considered above.

5.1.2 Disequilibrium in the money market and monetary growth cycles

So far, the rate of inflation p has been defined implicitly by money-marketequilibrium and has been derived from this equilibrium condition bymeans of appropriate calculations. Starting now from disequilibrium in themoneymarket serves (again) the purpose of presenting an explicit theory ofthe rate of inflation (whether a bad or good one has already been discussedin chapters 2—4). This explicit approach to the determination of the rate ofinflation yields (see section 2.2)

p��(m� h

�y� h

�(r�� ))� �� (1� �)(

�� n)

��(m� h

�f (l)� h

�(r��(l)��) )� �� (1� �)(

�� n).

Coupled with the laws of motion for l and � (see section 5.1.1.1), this nowgives rise to a three-dimensional dynamical system in the variables l,m, and� (m�

�� n� l1 � p by definition), which can be expressed as

l1 � n� s�(�(l)� g� )� (1� s

�)(

��)m, (5.6)

m� �(�� n��)� l1 � �

�(m� h

�f (l)� h

�(r�� (l)� �)), (5.7)

�� (��(m� h�f (l)� h

�(r��(l)��))

� (1� �)(�� n� �))��(�

� n��). (5.8)

In comparison to section 2.2 we have again one more dynamic law, i.e.,equation (5.6), which substitutes thereby the algebraic condition n�

s�(�(l)� g� )� (1� s

�)(

��)m of that earlier section. The two-dimen-

sional dynamical system of section 2.2 was shown to exhibit considerablepotential for Hopf bifurcation results by means of its various adjustmentspeed parameters. We expect here that this also holds true for the extendeddynamical system (5.6)—(5.8). We will demonstrate that this is indeed thecase by taking the parameter �� as a typical example of a bifurcationparameter. In order to do this, the coefficients of the characteristic poly-nomial of the dynamical system (5.6)—(5.8) have to be calculated first andthen used to form the conditions of the Routh—Hurwitz theorem.

Lemma 5.1: The determinant of the Jacobian of the dynamicalsystem (5.6)—(5.8) at the steady state is given by

det J�� l�m

�(�� )��[s�� a(1� s

�)n]


where a� h�f �(l

�)� h

��(l). This determinant is therefore in particular

negative when the Nagatani condition a� 0 (h�sufficiently small) is fulfil-

led.

Proof: The determinant det J is readily calculated as:

det J�m�l��

�s�� (1� s

�)n �(1� s

�)m

��a ��

��

�h�

0 0 ��

which implies the above assertions.�

Lemma 5.2: The principal minors J�,J

�, J

of J are given by

(a� 0 again)

J��m

� �(1� s

�)n �(1� s

�)m

�� (1��/��)

�� (��)� (�

�h�� )�� 0

if �� (1� s

�)n,

J�� l

��

�s�� (1� s

�)m

��a�

��(1� �)��/��

�h� � ,

J� l

�m

� ��s

�� (1� s

�)n

��a ��

� �� 0.

Proof: By straightforward calculations.�

Lemma 5.3: The trace of J is given by: �s��l

��

(�� (1� s

�)n)m

�� (1� �)�� h�

Proof: A straightforward calculation.�

Proposition 5.1: Assume a� 0 and �� (1� s

�)n. The steady

state of the dynamical system (5.6)—(5.8) is locally asymptotically stable forall ��,�� 0 that are chosen sufficiently small.

Proof: The Routh—Hurwitz conditions for local asymptotic sta-bility are: trace J� 0, det J� 0, J

��J

��J

� 0, and b� (� trace

J)(J�� J

��J

)� det J� 0.

It is obvious that trace J� 0 and J��J

��J

� 0 for �� 0 under the

given assumption �� (1� s

�)n. Furthermore det J� 0 for �� 0 and


Figure 5.1 The determination of the Hopf-bifurcation parameter

a� 0, and det J� 0 for �� 0 and a� 0. In the latter case, we howeverhave in addition that (� trace J)(J

��J

�� J

)�� det J� 0 must hold.

These Routh—Hurwitz conditions must therefore hold for all �� sufficient-ly small, since the involved expressions are all continuous functions of��.�


�)n. There exists a

unique parameter value �� such that b(��)� 0, and there is local asymp-totic stability as in proposition 5.1 for all �� (0, ��) and local instabilityfor �� . At �� the Jacobian J of (5.6)—(5.8) has one negative and a pairof pure imaginary roots.

Proof: It is easy to see that J�,J

�, J

, trace J and �det J are all

linear functions of type �� of �� with positive parameters �, �, up toJ(where �� 0), J

�(where � � 0), trace J (where �� 0), and det J (where

�� 0). The expression b� (� trace J)(J��J

��J

)� det J is therefore a

quadratic function of �� with b(0)� 0 and b� 0 at the uniquely deter-mined values ��, denoted �� and ��, where trace J (and J� �J

�� J

)


become zero. This gives rise to the graphical situation shown in figure 5.1.By virtue of the given information, there are therefore only two situ-

ations possible for the function b(��) (Sit. 1 or Sit. 2). Both of thesesituations imply that there is a unique parameter value �� 0 at which thesystem loses its stability, since b changes its sign (from a positive tonegative) when this value is passed (from the left). Furthermore, b cannotbecome positive again, before �� passes the value �� where the trace of Jbecomes positive, i.e., there is no return to local asymptotic stability.�


�)n. Then there

exists exactly one value �� of the parameter �� where the dynamicalsystem (5.6)—(5.8) undergoes the Hopf bifurcation by switching from localasymptotic stability to local instability as �� passes through �� frombelow. Generally, there is either the death of an unstable limit cycle or thebirth of a stable limit cycle as the bifurcation value �� is crossed frombelow.

Proof:Analogous to the proof in Benhabib andMiyao (1981).�

This proposition reformulates theorem 2 in Benhabib and Miyao’s (1981)treatment of the Tobin model for the Tobin model currently under con-sideration. It allows for the same comments as the original theorem ofBenhabib and Miyao and states in particular that there exist periodiccycles either at �� or in a neighborhood to the left (repelling) or to the right(attracting) of this critical value.Assuming an appropriate nonlinearity in the h

�component of the

money-demand function, we have shown in the corresponding case of fixedproportions (section 2.2) by means of the Poincare—Bendixson theoremthat there then also existed a limit cycle from the global point of view. Thisresult is easily transferred to the present case of a three-dimensionaldynamical systemwhen the condition s

�� 1 holds. In this case we get, from

(5.6)—(5.8), l1 � n� g� ��(l),��(l)� 0, and m, �� given as in section 2.3 with�� f (l)� f �(l)l in the place of �� n� g� .As the first dynamic law here shows, the steady-state value of l can only

be disturbed by real shocks (n, g� , f ( · ) ). The results of section 2.2 (for s�� 1)

therefore immediately apply to the above special case as long as � remainsat its steady-state value. The monetary sector thereby exhibits a furtherlimit cycle generatingmechanismwhich is due to a weakening of the Cagan(money-demand) effect far off the steady state. We expect this limit cycleresult to hold also for real disturbances and resulting adjustments of theratio l to its new steady-state value as well as for savings ratios s

�� 1 (at

least) sufficiently close to 1, in which case the dynamical equations for l, m,


and � are interdependent due to the Tobin effect then present.

5.1.3 Steady-state analysis and the enlarged Tobin effect

So far we have considered stability issues in their resemblance to thoseobtained for the case of fixed proportions in production, and neglected thequestion of the existence and uniqueness of steady-state solutions and theirproperties under smooth factor substitution. This gap needs to be closed, inparticular since the interest in models of the Tobin type often is concen-trated on the question of (non-)superneutrality of money and thus onproperties of its steady state.The steady state of the models of this section is determined by the

following equations

��

�� n, (5.9)

n� s�(�(l

�)� g� )� (1� s

�)nm

�, (5.10)

m�� h

�f (l

�)� h

�(r� ��(l

�)�

�� n), (5.11)

where r� is now an arbitrary positive parameter, not necessarily equal tor��(l

�)�

�� n as we have hitherto assumed for simplicity. Equations

(5.10) and (5.11) give rise to the equation

const.� n� s�g� � (1� s

�)nh

�(r� �

�� n)

� s��(l

�)� (1� s

�)n[h

��(l

�)� h

�f (l

�)]� s

��(l

�)� b(l

�),

for the unknown full employment labor intensity l�. Since r� �

�� n

should hold true, the constant on the left hand side of this equation can beassumed to be positive. Furthermore, we again assume that the Nagatanicondition m

�� 0 holds true for the above money-demand function m�

h�f (l)� h

�(r� � �(l)�

�� n).� The function on the right hand side of the

above equation is then strictly increasing in both of its components (and 0for l� 0, also with respect to both components). The Inada conditions forthe production function y� f (l) then imply that there is a (uniquely deter-mined) value of l

�for which the above equation is fulfilled.

Figure 5.2 depicts this finding. It also shows that the steady-state value ofthe labor intensity ratio l

�depends negatively on the growth rate

�of the

money supply, which is the central result (on non-superneutrality of themoney-supply process) of models of the Tobin type.Increasing the growth rate

�increases the steady-state rate of inflation

��

�� n by the same amount, thereby decreasing the return of holding

money as a financial asset. The relative composition of assets will thereby

� h�� 0, for example, implies this condition.


Figure 5.2 The non-superneutrality of money

be shifted towards capital, increasing capital intensity k�K/L and outputper worker x�Y/L, while decreasing at the same time labor intensity l andoutput per unit of capital y. This summarizes the basic quantity effects ofchanging money supply growth which, under smooth factor substitution,gives rise not only to distributional effects (as in section 2.1), but also, dueto the neoclassical theory of income distribution, to changes in the ratio inwhich the two factors of production are employed.We make two final remarks in respect of the Tobin model. Firstly, the

assumption of smooth factor substitution becomes a necessary one in thecase (common in the literature) where s� s

�� s

�is assumed, since there

will be no steady-state solution under fixed proportions with a uniformrate of savings out of profit and wages.Secondly, on the basis of the Nagatani condition m

�� 0, we have that

the Tobin general equilibriummodel type is globally asymptotically stableas is the Solow real growth model l1 � n� s

�(�(l)� g� )� (1� s

�)(

�� )m,

as long as expectations � are fixed (�� n, for example). The usually

observed instability of the Tobin-type dynamics therefore comes from


quite another source, as can, for example, be seen by removing the Tobineffect from this monetary growth model type, by setting s

�� 1. In the case

of adaptive expectations and a somewhat sluggish adjustment of prices weget in this case from lemma 5.3 the stability condition

0� trace J�� (1� �)��[�� 1/h

�]h

�m

�,

which once again shows the stabilizing potential of the parameters �� and�� 1, while the interaction of adaptive expectations and the adjustmentspeed of prices is a potential source of instability. This, however, should berelated to mechanisms investigated in particular by Cagan and not somuch with studies of the Tobin effect and the stable dynamics impliedwhen it is working in isolation.

5.2 The Keynes–Wicksell case: increased stability through increasedflexibility

The extension of the general Keynes—Wicksell model (3.1)—(3.24) and itsintensive form dynamical laws (3.25)—(3.29) to smooth factor substitution isparticularly simple and straightforward. In the place of the equations in(3.7) we now assume that employment L is determined by the marginalproductivity principle � �F

�(K,L)�F

�(1, l), l�L/K, and that the

corresponding potential outputY��Y is given by the production functionY�F(K,L) underlying this marginal productivity rule (V�L/L again).This reformulation of the employment and output relationship is all that isneeded to extend the Keynes—Wicksell model to continuous substitution.There is no change in the formal presentation of the laws of motion

(3.25)—(3.29) of this model. In the accompanying algebraic equations, how-ever, we obtain, due to the above considerations,

l� l(�)� f��(�), y� y(l)[�(�)� f (l(�) )�� · l(�)� �],

in place of the hitherto fixed coefficients l and y (the parameter x�Y/L isno longer needed).In the equations for the steady-state values (3.30)—(3.35), we have now to

replace the first condition (3.30) by

y�� f (l

�), l

�� l(�

�), l

�� l

�/V�

which makes the calculation of the steady-state values for l, y, and �slightly more interdependent. In the case

��

�(b

�� 0), the basic equa-

The functions l1 (m) and m(�) are the basis of the Tobin effect (the first is increasing, the

second decreasing). Note that � is given at each moment of time in this model type, so that there is the causalstatic relationship l� l(�) in this context.


tion for the determination of these values reads ��(�

�)�

(n��h�f (l(�

�)))/s

�(1� �) , for which a unique solution �

�� 0 has to be

shown to exist. This is again easy if h�� 0 is assumed.

The general five-dimensional dynamical system governing the dynamicsof this Keynes—Wicksell model is again reduced to a four-dimensionaldynamical system by the assumption t�� const., and to a three-dimen-sional one by way of the further assumption ��

�� n (a consequence of

assuming �� ). The basic case of an independent real sector of theeconomy is finally obtained by assuming h

��, i.e., r� r

�,

�� 0. The

following investigations will start from this basic case and briefly considerhow the propositions of the fixed proportions situation are changed by theassumption of smooth factor substitution.

5.2.1 The Goodwin case

The two differential equations

u� ��(V�V� ), u��/x��l/y,

V1 � i(y� �� uy� r��

�� n),V� l/l,

of the fixed proportions case of section 3.4 now demand the use of thedynamic variable � in the place of u, since l and y now both depend on �.This variable change allows again a treatment of this dynamical system bymeans of a suitably chosen Liapunov function H.In order to show this, we first calculate the percentage rate of change of

the variable V, making use of the functional relationship l� l(�) (or�� f �(l) ) to obtain

V1 � l1 � l1 � l�(�)�� /l(�)� n�K1 � �(�)�� g(�).

In this last expression, �(�)� 0 stands for l�(�)�/l(�) and g(�), (g�(�)� 0)expresses that K1 depends (only) on � in a positive way, and thus on � in anegative way,� independently of whether it is explained by I/K or S/K (or acombination of both).The above two differential equations thus give rise to

�� (V�V� )� h(V ), (5.12)

V1 � �(�)��(V�V� )� g(�), (5.13)

in the case of smooth factor substitution. This reformulation of the modelallows again for the type of Liapunov function we have used in section 3.3for (3.29) and (3.30), which here reads

� To see this note that �(�)� f (l(�))��l(�)� � ��(�)� � l(�)� 0.


H(�,V )��

��

g(�& )/�& d�& ��'

'

h(V� )/V� dV� .

It is easy to show that this function fulfills

H� � �(�)h(V )�� 0, if V�V� �V�(� 0 otherwise).

Hence we obtain the following.

Proposition 5.4: The steady state ��[g(�

�)� 0],V

��V� of the

dynamical system (5.12)—(5.13) is globally asymptotically stable in thepositive orthant of R�.

Proof:A standard application of the theory of Liapunov functions(see, e.g., Hirsch and Smale 1974, pp.195—6).�

Smooth factor substitution thereforemakes the center type dynamics of theoriginal Goodwin growth cycle model a convergent one. In addition it canbe shown that this convergence becomes monotonic (loses its generallycyclical properties) if the elasticity of factor substitution is chosen suffi-ciently high. By contrast, the originally closed orbit structure of the Good-win model is approached again, if this substitution elasticity gets smallerand smaller (the same holds if the parameter �

�is decreased instead). These

considerations indicate how the models we considered in section 3.3 will bemodified in the presence of a neoclassical production function Y�

F(K,L).

5.2.2 The Rose case

We recall that the Rose case is obtained by taking all the assumptions thatgive rise to the Goodwin case, but replacing

�� 1 by

�� 1. Equation

(5.12) is in this case thus expanded to the form

�� [(1� �)��(V�V� )� (

�� 1)�

�(i( · )� n� s( · ))], (5.14)

where i( · )� n� i(�(�)� r��

�� n)� n and s( · )� s

�(�(�)� t�) are the

two functions we have denoted by g(�), l(�) in the preceding subsection. Asfar as the dynamic law (5.13) is concerned, it is now more appropriate tomake use of a formulation in terms of l(� l(�)/V), which turns out to be

l1 � n� g(�), g(�)� i( · )� n or g(�)� s( · ) (5.15)

This, of course, requires that V be replaced by l(�)/l in equation (5.14).In order to derive global stability (but generally not asymptotic stability)

for this extended dynamical system of the real sector, we assume again for


Figure 5.3 A restricted neoclassical production function

the ��(V�V� ) part of the money-wage Phillips curve of this model the

nonlinear shape we have used in section 3.4 in order to make the dynamicsof the Rose extension of the employment cycle viable. The nonlinearinvestment function that we also used in section 3.4 is now no longernecessary in order to obtain such a result, and is thus replaced here by anonlinear smooth production technology.Let a and b(0� a� b) again denote the limits of the assumed nonlinear

Phillips curve, where wage inflation becomes �� and ��, respectively(see page 149, figure 3.2). To ease the following presentation, we assumehere for simplicity that the production function y� f (l) is of the formdisplayed in figure 5.3. This form has the property that the marginalproduct of labor is less than infinite at l� 0, and becomes zero at a certainfinite level of the labor intensity l.In this case, the dynamical system (5.14)—(5.15) gives rise to the phase

diagram shown in figure 5.4. Due to the above assumption on the range ofpossible marginal products of labor, we know that the decreasing functionl(�) must cut the axes of the positive orthant at �

��and l

��. This fact

then also holds true for the rescaled functions l� l(�)/a (where ��


Figure 5.4 The viability domain of the Rose dynamics under smooth factorsubstitution

holds, since V� l(�)/l� a along this curve) and l� l(�)/b (where�� holds true). Note here that these two curves have the same originB and that they enclose the open domain of (�, l) combinations where thedynamics (5.14)—(5.15) is well-defined (the open region ABC).The isoclines �� 0, l� � 0 of (5.14) and (5.15) are given by

l� l(�)��

(1� �)��(i( · )� n� s( · ) )

1� �

��V� ��,

��.

Due to the assumed shape of the ��function (see again figure 3.2), we know

that the first expression is always well defined and must always lie betweenl(�)/b and l(�)/a (see figure 5.4). The above two isoclines then divide thephase space as shown in figure 5.4.By employing a proof similar in spirit to that in Benassy (1986a) (see also

Flaschel and Sethi 1996), it is not very difficult to show that the Poincare—Bendixson theorem can be applied to the situation depicted in figure 5.4 ifwage flexibility is so low at the steady state that the �

�term implies its local


instability (i� s�). Sluggish wages near the steady state, and ever-increasing

wage flexibility far off it, therefore imply the existence of a limit cycle as inRose (1967), now independently of any nonlinearity in the market forgoods.The details of such an analysis are spelt out in Benassy (1986a) and

Flaschel and Sethi (1996) and will not be repeated here. Here it is sufficientto have sketched the result, that in Keynes—Wicksell models (of a specialtype) we can get global stability solely through appropriate assumptionson wage flexibility in conjunction with smooth factor substitution. Notehere that the viability domain ABC of this dynamical system can betailored to empirically plausible magnitudes simply by making an appro-priate choice of the two bounds a and b for the Phillips curve. Note also,however, that this neoclassical supply-side dynamical behavior (which canbe much more complicated than is generally admitted by supply-sideeconomists) is not yet too convincing from an empirical point of view.Indeed, this comment is generally true for theKeynes—Wicksellmodel type,as we have remarked in earlier chapters.

5.2.3 Monetary growth dynamics

Allowing now again for interest rate flexibility (h��) extends the dy-

namics (5.14)—(5.15) to a three-dimensional dynamical system by means ofthe extended relationships

i( · )� n� i(�(�)� r� �� n)� n,

s( · )� s�(�(�)� t�)�

�m,

r� r�� (h

�y(�)�m)/h

�,

and the further dynamical law for m(� (M/pK))

m� �� g( · )� �� (p��), ��

�� n,

where g( · ) is again given by either i( · )� n or s( · ) and

p�� [��(i( · )� n� s( · )�

��(V�V� )],

V� l(�)/l,y(�)� f (l(�)).

Note first that the function �(�)� y(�)� ��l(�) is but a nonlinearextension of the formerly linear expression used in section 3.5 (see section3.2 for the definition of �), namely �(�)� y� ��l, l� y/x, y,x� const. (��(�)�� l( · )� 0 in both cases). Note also that the rateof interest now also depends on�, so that r� � h

�f �(l)l�(�)� 0 now holds.

This negative dependence on the real wage can be aggregated with thenegative dependence of � on�, so that neither the i( · ) nor the s( · ) equationis changed qualitatively by the addition of smooth factor substitution. This


fact extends to the p�� equation as far as its ��component is concerned.

The ��component, however, is here (and also in the �-equation) different

from the situation of fixed proportions, where it was a function of l solely(l� y/x� const.). Now l� l(�) due to smooth substitution which intro-duces via �

�( · ) a new derivative into the determination of the Jacobian of

such a three-dimensional dynamical system (in the � � as well as in them� part of it).The sign structure of the Jacobian at the steady state of section 3.5 was

J��? � �

� 0 �

? � �� ,and it is now augmented by the addition of a further matrix of the form

�� 0 0

0 0 0

� 0 0� .Such an addition does not modify the results we have stated in section 3.5for the matrix J as far as its trace, the principal minors J

�, J

�, and J

, and

the determinant of J are concerned. This conclusion can also be extendedto the final Routh—Hurwitz condition

b� (� trace J)(J��J

��J

)�det J,

which then shows that the addition of smooth factor substitution increasesthe range for local asymptotic stability (see proposition 3.6 in section 3.5 inparticular).Despite this improvement in the stability properties caused by smooth

factor substitution, the qualitative stability results of section 3.5 are notsignificantly modified by this flexibility in the use of factors of production.With this observationwe close our reconsideration of the Keynes—Wicksellprototype and its (by and large confirmed) stability properties.

5.3 The Keynesian case with smooth factor substitution

5.3.1 The Keynesian IS—LM growth model generalized

This section presents a possible, though in its particular form not toocompelling, extension of the production function used in the Keynesianmodel of chapter 4, namely, to neoclassical smooth factor substitution aswas done for the Tobin and Keynes—Wicksell model types in sections 5.1and 5.2. It is here included for methodological reasons and is intended to


show that the standard neoclassical production function can be used in theKeynesian context as well, without leading to a loss in the substance of thisunique approach to underutilized (or overutilized) labor as well as un-derutilized (or overutilized) capital.The Tobin and Keynes—Wicksell type models of monetary growth have

generally been associated with the assumption of a neoclassical productionfunction and have been analyzed within such a framework. To do the samefor ourKeynesian typemodelmay not appear as natural as for these earliersupply-side models, since there has been (and perhaps still is) a tradition inKeynesian growth economics which rejected (and continues to reject) theuse of such a production function, having its roots in the works of Harrod,Kaldor, and Sraffa in particular. Yet, we believe (see also Marglin, 1984a,for similar arguments) that the particular type of production theory em-ployed should not prevent the formulation of a proper Keynesian modelwhere both labor and capital can be unemployed (or overemployed).Keynesian monetary growth dynamics should also exist as a meaningfulmodel if technology is of the neoclassical production function type, quiteindependently of the question whether this function is really appropriate inthis context. Assumptions about technology should consequently not bedecisive for a proper understanding of various schools of thought and theirway of formulating the essential aspects of the forces governing monetarygrowth and cycles.Therefore, our Keynesianmodel of chapter 4, with its distinctive features

regarding the determination and implications of underutilized capital (be-sides underutilized labor), should survive the introduction of a neoclassicalproduction function in place of the fixed proportions technology so farassumed.We shall make use of such a production function as it is described in the

introductory pages of Sargent (1987, ch. 1) for the so-called ‘‘Classical’’model: ‘‘In this one-good economy capital represents the accumulatedstock of the one good that is available to assist in production. We assumethat at any moment the stock of capital is fixed both to the economy and toeach individual firm’’.�In light of the foregoing we shall now proceed as follows. Since prices

and wages are given at any moment in time in our Keynesian model type,and since output and interest are determined by the IS—LM part of themodel, firms cannot adjust their rate of capacity utilization to the profit-maximizing full capacity level, since they may then experience unintendedinventory changes in particular. Such changes will be treated in chapter 6,but here we insist on IS—LM (and thus goods-market) equilibrium, which

� See Sargent (1987, pp.8ff.) for further details of such a description.


means that firms just produce the level of output that equals effectivedemand at each moment in time. As in other fixed-price approaches toKeynesian dynamics, we consequently must assume here that firms areuniformly rationed (in proportion to their profit-maximizing capacitylevel) in order to guarantee that the allocation of labor in our one-goodeconomy is still an efficient one, and that their output is therefore indeeddescribed by the value of the macroeconomic production function at thecurrent level of the capital stock and of employment. In Sargent’s (1987,pp.7ff.) analysis of the allocation of labor among firms which differ in size(but not in technology), this is achieved by the use of the marginal produc-tivity rule for each individual firm. As a consequence, output is always at itsprofit maximizing or full capacity level, which is appropriate in a Keynes—Wicksell model, but not in a Keynesian one as we have shown whencomparing the model types of chapters 3 and 4. We here therefore assume(by the above rationing hypothesis) that labor is always used efficiently, butgenerally not at the full capacity level of firms. Of course, an inefficient useof the employed may indeed occur, and will complicate the analysis con-siderably, since there is then a wide variety of choices for the rationingscheme to be employed with respect to the given number of firms.Here, however, firms are rationed in proportion to their profit-maximiz-

ing productive capacity (they thus exhibit a uniform rate of capacityutilization with respect to this level) and they increase (consciously orunconsciously, by the law of demand) this capacity by price increases(thereby making higher output levels profitable), or quantity reactions(intended and realized increases in the capital stock), whenever the usage oftheir profitable capacity passes a certain thresholdU� � 1 (as in chapter 4).It is assumed in this chapter that firms may approach their full capacity

utilization level U� 1, but that they never reach this point of (from theviewpoint of profitability) ‘‘absolute’’ capacity utilization where marginalwage costs have become equal to the momentarily given price of output. Itis, however, easily conceivable (see Powell and Murphy 1997 for a macro-econometric approach of this type) that firms extend their productionbeyond their profit maximum in order to satisfy their customers, in orderto defend their market shares and the like. Thus this upper economicbound is in fact not so binding as it is generally believed to be in theneo-Keynesian description of rationing equilibria of Classical type.Smooth factor substitution, therefore, in principle allows the capital

stock constraint to be ignored (at least temporarily during the businesscycle). It is, therefore, in fact much easier to treat than the case of fixedproportions in technology, whereU� 1 is an absolute limit to any further

At least not at the present stage of the formulation of the model.


increase in the level of production. Instead of a switch into the ‘‘Classicalregime’’ of neo-Keynesian type we may therefore simply allow for a signifi-cant increase in the adjustment speeds of prices and the rate of investmentwhen the level of (profit maximizing) potential output is reached andcrossed, which, as Chiarella et al. (1999) show, is then occurring only overshort intervals of time. The present chapter can therefore also be inter-preted, as in Powell and Murphy (1997), as providing a theory of adjust-ments of the price level and of the capital stock based on medium-runoriented first-order conditions, which act like moving targets for these twodecision problems of firms, while the decision on the rate of production isbased on (correct) point expectations on the sales of firms.Let us stress that the usage that is here made of the concept of a

neoclassical production function is not very close to the putty—putty inter-pretation in the literature on this function. Rather it is here only assumedthat the various capital stocks of the firms of this economy are utilized in anoverall efficient way by means of a specific allocation of labor throughoutthe economy. This description of actual output and efficient labor alloca-tion in conjunction with the above concept of profit-maximizing potentialoutput is all that is needed to go from our Keynesian model of chapter 4 tothe case of ‘‘smooth factor substitution’’ in production, a technology con-cept which here thus only allows for such substitution in the medium andthe long run.

The equations of the Keynesianmodel with smooth factor substitution are:


��w/p, u��/x,� � (Y� �K��L)/K, (5.16)


%� 1. (5.17)



�pY� h

�pK(r� � r), (5.18)

C� �L� (1� s�)[�K� rB/p�T ], s

�� 0, (5.19)

S��L�Y�


� s�[�K� rB/p�T ]� s

�Y��

� (M� �B� � pE� )/p, (5.20)

L1 � n� const. (5.21)


��F�(K,L�),Y��F(K,L�),U�Y/Y�, (5.22)

L det’d via Y�F(K,L),V�L/L (5.23)


I� i�(�� r��)K� i

�(U�U� )K� K, � n, (5.24)

pE� /p� I, (5.25)

K1 � I/K. (5.26)


T� t�K� rB/p, t�� const., (5.27)

G�T� rB/p��M/p, (5.28)


M1 ��, (5.30)

B� � pG� rB� pT�M� [� (��

�)M]. (5.31)



�pK(r� � r)[B�B,E�E], (5.32)

pE��pK/(r��), (5.33)

M� �M,B� �B� [E� �E� ]. (5.34)


S� pE� � S

��S

��Y� �K�C�G� I� p

E� . (5.36)


w� ��(V�V� )�

�p� (1�

�)�, (5.37)

p��(U�U� )�

�w� (1�

�)�, (5.38)

�� (p��)��(�� n� �). (5.39)

The range of changes in the formulation of the Keynesian monetarygrowth dynamics of chapter 4 is given by the equations (5.22) and (5.23).The equations in (5.22) describe the potential employment schedule (whichis based on the marginal productivity rule � �F

�) and the corresponding

concept of potential outputY�. Actual employment is derived in (5.23) fromactual output (governed by the principle of effective demand), and thesetwo actual magnitudes are used in the same way as in chapter 4 to definethe actual rate of capacity utilization U and of employment V. The thusdefined rate U is then used as in that chapter to explain the demandpressure components i

�( · ), �

�( · ) in investment and pricing behavior (see

equations (5.24) and (5.38); the cost-push components are as before).Summing up, the model of chapter 4 is thereby changed in a minimal

way without giving up any of its essential principles of the working of a


Figure 5.5 Potential and actual employment and output

capitalist economy. By contrast, the literature on Keynesian dynamicsgenerally identifies the ratios U and U� with 1 and introduces thereby aconsiderable element of confusion into the discussion of Keynesian dy-namics, as we shall further explain in the next section. This is here avoidedby the simple, and at present admittedly restrictive, procedure (and as-sumptions) that firms always operate to the left of the value L� in the figure5.5,� and that they speed up price increases and capacity extension beyondthe threshold L� (corresponding to the NAIRU level U� ) well before fullcapacity utilizationU� 1 is in fact reached.� In this way, it may be possiblethat the economy evolves according to these (and further) rules withouthitting the ceiling L� where a regime switch could take place. The analysisof such regime switches could be considered as an extension of the present

� Note here that an increase in the real wage � implies an increase in the rate of capacityutilization U for any given level of employment L — due to a closer profitability barrier.

� A more general behavior of firms with respect to this ‘‘capacity limit’’ is motivated andmodeled in Chiarella et al. (1998), based on the empirical observation that firms in factchoose to produce (from time to time) beyond the point where output price equals marginalcosts in order to satisfy their customers’ demand.


model.�� This is in our view the simplest possibility for the extension of theKeynesianmodel of chapter 4 to variable factor proportions in technology.The abovemodel gives rise to the following set of intensive form differen-

tial equations�� in the variables �, l, m, and �:

�� [(1� �)��(l/l�V� )� (

�� 1)�

�( f (l)/y�(�)�U� )], (5.40)

l1 � n� s�( f (l)� � ��l� t�)�

�m, (5.41)

m� �� n� [�

�( f (l)/y�(�)�U� )

� ��(l/l�V� )]� l1 , (5.42)

�� [��( f (l)/y�(�)�U� )�

��(l/l�V� )]

� ��(�� n� �), (5.43)

where l is given by the solution of

i� �f (l)� ��l� t�� r

��h�f (l)�m

h�

�� i

�( f (l)/y�(�)�U� )� n� s

�( f (l)� � ��l� t�)�

�m, (5.44)

and where y�(�)� f (( f �)��(�)) is the full capacity output—capital ratio offirms at the prevailing level of the real wage �. Note here that (y�)�(�)� 0holds because of the properties of the production function Y�F(K,L).Differentiating equation (5.44) with respect to the statically and dynami-

cally endogenous variables gives the expression

[(s�� i

�)( f �(l)��)� i

�f �(l)/y�(�)� i

�h�f �(l)/h

�]dl

� [(s�� i

�)l� i

�f (l)(y�)�(�)/(y�(�))�]d�

� (�� i

�/h

�)dm� i

�d�.

We assume that the bracketed expression in front of dl is nonzero at thesteady-state solution of the model (and thus in a neighborhood of it; see thefollowing presentation of these steady-state values). There thus exists aunique function l� l(�,m,�) in a neighborhood of this steady state whichjust solves the above equilibrium condition (5.44) and which is continuous-ly differentiable.Note here that the special conditions

�� 0, h

��, and ��

�(�� n) again imply that l is a function of only the real wage �

which can be discussed in the same way as the one we considered in section4.2 in the case of fixed proportions. The above differential of the equilib-rium condition then implies

�� See Chiarella et al. (1999, ch. 3) for the details on treating such ceilings.�� Compare the corresponding equations in section 4.1.2.


dl

d��(s�� i

�)l� i

�f (l)(y�)�(�)/(y�(�))�

(s�� i

�)( f �(l)��)� i

�f �(l)/y�(�)

,

which is easily reduced to the corresponding expression in section 4.2 onthe basis of the assumptions y� f (l)�xl, y�� const.(u��/x) of thatsection.�� There will therefore again exist three possible situations for theimplicitly defined function l(�) as in section 4.2. Note here also that therange of validity of case 3 of the l(�) relationship is enlarged by theinclusion of smooth factor substitution due to the additional and negativeexpression �i

�f (l)(y�)�(�)/(y�(�))� that now appears in the numerator of

the preceding fraction.We do not go into the details of a treatment of the resulting two-

dimensional dynamical system here, but do note that there is now just oneterm to be added to the Jacobian we considered in section 4.3, which isgiven by (1�

�)��f (l)(y�)�(�)/(y�(�))�, and which appears in its J

��entry.

This term is negative for �� 1 (due to (y�)�(�)� 0), and it adds consider-

able further stability to this two-dimensional case� independently of thesign and strength of the term (l)�(�), and thus of the particular of the threecases of section 4.4 that may be under consideration.There is again a unique steady-state configuration for the dynamical

system (5.40)—(5.43), with ��, l�,m� 0 when the usual assumptions on the

neoclassical production function y� f (l) are made and when the par-ameters

�and h

�are chosen sufficiently small (such a choice can be

justified by reference to the empirical magnitudes of these parameters). Inthe present case t�� const. (r� � r

�as always) this steady state is given by�

n� s�( f (l

�)� � ��

�l�� t�)�

�m

�,

f (l�)�U� y�(�

�)�U� f (( f �)��(�

�) ),

m�� h

�y�,

y�� f (l

�), l

�� l

�/V�

��

�� n,

�� y

��

�l�,

r��

��

�� n.

The existence of a unique steady state is easily proven by noting that thesecond of the above steady-state conditions defines a falling function l(�)via the implicit function theorem, while the first one implies by the sametheorem a rising function l(�) when the third steady-state condition is

�� Since we then get from the above thatdl

d��

(s�� i

�)l

(s�� i

�)(x��)y�� i

�xy�.

� The determinant is not changed in its sign by this single additional term, since J��

� 0 stillholds.

� Due to the same reasoning as used in section 3.2 for such a steady-state determination.


inserted into it (and when the parameters �and h

�are chosen sufficiently

small��). Having determined the steady-state values of � and l, it is thenobvious how to get the steady state values m

�, l

�, y

�, �

�, and r

�(the

steady-state value ��is determined independently of all these calculations).

Note that the steady-state value of r has been used in the definition of themoney demand function, by which it is in fact established that the abovetype of steady-state calculation is possible. As in the preceding chapter, weassume that the parameters of the model are chosen such that the threesteady-state values for �, i, and m are all positive.The analysis in section 4.5 of the basic case of an integrated monetary

growth dynamics is obviously also relevant for the present case of smoothfactor substitution if we artificially assume in the formulation of the fourdifferential equations (5.40)—(5.43), and only there, that potential outputper capital y� is a fixed magnitude (in place of its negative dependence onthe real wage�). The steady-state values and the properties of the functionl(�,m,�) that is determined by IS—LM equilibrium as shown above willgenerally differ from each other for the case of fixed factor proportions andsmooth factor substitution. But the Jacobian in this artificial situation isthen of the same form as in section 4.5, and thus gives rise to the samequalitative conclusion as we derived there. Essential differences to thisearlier consideration of local stability and Hopf bifurcation situations cantherefore only arise from the additional elements in this Jacobian that stemfrom the dependence of y� on the real wage�. These additional expressionsgive rise to the matrix

J(

�� (

�� 1)�

�U�� 0 0 0

0 0 0 0

� ��U�m 0 0 0

�� U� 0 0 0��

� 0 0 0

0 0 0 0

� 0 0 0

� 0 0 0� ,which has to be added to the one in section 4.5 in order to give the fullJacobian matrix J of the system (5.40)—(5.43) at the steady state. Here U�denotes the positive dependence of the rate of capacity utilization on thereal wage and is given by �f (l)(y�)�(�)/(y�(�))�.Making use of this particular qualitative structure of additional entries

in the Jacobian J of the dynamics at the steady state, it is then easily shownin reference to the calculations and assumptions of section 4.5 that the traceand the determinant of this matrix J become more negative through theinclusion of these additional terms. At the same time the principalminor J

�� Wehave f �(l

�)��

�by the chosen side conditionU� 1 (motivated through the choice of a

steady state utilization rateU� which is less than one). The positive slope of the consideredfunction then follows if it is assumed in addition that f �(l

�)(1�

�h�)��

�holds true.


becomesmore positive thereby (the other principal minors are not changedthrough the inclusion of J

(). These results are in particular due to the

property J��

� 0 of the Jacobian J. Three of the four Routh—Hurwitzconditions for local asymptotic stability are therefore improved throughthe inclusion of J

(. The remaining condition, �trace J · (J

��

J��J

)�det J� 0, however, shows no clear-cut change since det J� 0

has become more negative and since there is one term here added to itwhich is not cancelled by the additional expressions in �traceJ · (J

��J

��J

). The stabilizing potential of smooth factor substitution

consequently remains somewhat limited in the Keynesian prototypemodel.The general conclusion from this discussion nevertheless is that the

replacement of a fixed proportions technology with a neoclassical produc-tion function is of secondary importance in the evaluation of the basicproperties of proper Keynesian models of monetary growth with bothunder- or overutilized labor and capital. Here the case of fixed proportions(which we have treated in detail in chapter 4) has the twofold advantage ofcontaining already all necessary elements for the study of such a monetarygrowth dynamics and of allowing us to probe the central features of suchgrowth dynamics in a more transparent and less intertwined way. Thatsuch a simplification in the treatment of technology can indeed be illumina-ting will become particularly clear in section 5.3.2 where the (textbook)treatment of Keynesian dynamics that is currently the standard one isshown to be both misplaced and confusing as far as the Keynesian charac-ter of such standard models is concerned.

5.3.2 The bastard limit case: AD—AS growth

In textbook presentations of the so-called Keynesian static or dynamicmodel it is generally assumed that prices are completely flexible and thatthey fully adjust at each moment of time to marginal wage costs, which arein these models determined in reference to a neoclassical production func-tion. Nominal wages, by contrast, are in these formulations of Keynesianstatics or dynamics given at eachmoment of time, while their rate of changeis made endogenous by way of a standard money-wage Phillips curve (justas in the preceding chapters of this book).Allowing for complete price level flexibility (i.e. by setting �

��) in the

Keynesian model of the preceding subsection at first sight appears to beunproblematic, since this assumption has been the standard description ofaggregate supply in the Keynesian literature and since also Keynes (1936)did not refute this postulate of Classical economics (as he called thisassumption). Yet, looking at the consequences of making this assumption


in the context of the model of the preceding subsection shows that thismodel becomes a supply-side determined model thereby as far as utiliz-ation of the capital stock is concerned.Assuming �

�� first of all gives rise to the identity U�U� in place of

the price adjustment rule (5.38). Capacity utilization problems thereforedisappear from this form of the model, which makes it admissible to setU� � 1 in addition, as is customary. On the basis of given money wages, wethen get the conventional schedules of aggregate demand and aggregatesupply from the IS—LM equations (5.32) and (5.36), and the marginalproductivity principle (5.22). Keynes (1936) interpreted this principle asdetermination of the price level (p�w/F

�), supplementing his determina-

tion of firms’ output and the nominal rate of interest via multiplier theory(effective demand) and liquidity preference (IS—LM). Demand-constrainedproducers therefore were supposed to choose this level of prices to accom-pany their demand-constrained output decision.The formal consequence of assuming �

�� is, however, independent

of this reinterpretation of the marginal productivity rule. Output, interest,and prices are now simultaneously determined by the intersection of theaggregate demand and supply schedule as components of the temporaryequilibrium part of the model. Prices are no longer dynamically en-dogenous as in the preceding sections of this chapter, but have becomestatically endogenous. Nevertheless, their time rate of change has to becalculated (these become a complex expression of the dynamically en-dogenous variables as shown by Franke 1992a) in the application of theexpectations formation mechanism (5.39).Further straightforward consequences of this type of price flexibility are

that the i�term can now be ignored in the investment function (5.24).

Furthermore, it is customary in the presentation of the resulting model toset the values of

�and �� equal to 0.

The resulting model is then given by the following set of equations:


��w/p, u��/x,� � (Y� �K� �L)/K, (5.45)


%� 1. (5.46)



�pY� h

�pK(r� � r), (5.47)

C��L� (1� s�)[�K� rB/p�T ], s

�� 0, (5.48)

S��L�Y�



� s�[�K� rB/p�T ]� s

�Y��

� (M� �B� � pE� )/p, (5.49)

L1 � n� const. (5.50)


��F�(K,L),Y��F(K,L), (5.51)

V�L/L, (5.52)

I� i(�� r��)K� K� K, � n, (5.53)

pE� /p� I, (5.54)

K1 � I/K. (5.55)


T� t�K� rB/p, t�� const., (5.56)

G�T� rB/p��M/p, (5.57)


M1 ��, (5.59)

B� � pG� rB� pT�M� [� (��

�)M]. (5.60)



�pK(r� � r)[B�B,E�E], (5.61)

pE��pK/(r��), (5.62)

M� �M� ,B� �B� [E� �E� ]. (5.63)


S� pE� � S

��S

��Y� �K�C�G� I� p

E� . (5.64)


w� ��(V�V� )��, (5.65)

�� (p� �). (5.66)

This is the model that Sargent (1987) uses as basis for his analysis ofKeynesian dynamics.��The first thing that has to be noted here is that the same model is also

�� Coupled with the assumption s�� s

�� s on the savings propensities of capitalists and

workers, since this assumption does now, due to assumption of smooth factor substitution,allow for a steady-state solution.


obtained from the Keynes—Wicksell model (with smooth factor substitu-tion) by setting �

�� there. This assumption also suppresses the explicit

theory of inflation of this model by way of its implication thatI�S(U�U� � 1 is assumed in this supply-side oriented Keynes—Wicksellprototype model right from the start). Assuming �

�� therefore implies

in each case the missing causality of the following two equalitiesI�S,U�U� � 1 for both the Keynesian and the Keynes—Wicksell modeland makes them formally indistinguishable thereby.� Note here, however, that the marginal productivity principle � �F

�was explicitly used in the Keynes—Wicksell model (in the case �

��) to

determine employment through the there assumed behavior of firms asprice-taking profit-maximizing units which face no demand constraint.The same principle is used in our Keynesian model of the precedingsubsection (�

�� ) to determine the extreme (� profit-maximizing) level

of output Y�(Y��F(K,L�),F�(K,L�)��). This hypothetical level is then

compared with the actual level of output of demand-constrained, price-setting firms in order to formulate on this basis the effects on the rate ofinvestment and on the rate of change of the price level in view of thediscrepancy of actual output from a certain ‘‘normal’’ portion of thisprofit-maximizing output (in the framework of price-setting firms).In the limit �

��, this difference in the setup of causal relationships of

the Keynes—Wicksell and the Keynesian model type disappears and isreplaced by the conventional aggregate demand and aggregate supplyargument (determining the variables Y, r, and p), as we have discussedabove. In this limit, it is therefore no longer obvious whether employmentcan be basically imagined to be determined from the side of supply or fromthe side of demand. In fact it is determined by the intersection of aggregatesupply and aggregate demand, and thus from both sides simultaneously.Nevertheless, Keynesian theory is, or attempts to be, a theory of effectivedemand constraints on the market for goods with consequences for bothlabor and capital. One therefore has to judge to what extent it is stillrepresented in this framework of an infinitely flexible price level (and asluggishly evolving level of nominal wages).The non-Keynesian character of the present limit case is most strikingly

revealed when the case of fixed proportions in production is again as-sumed. We then get from the assumption �

�� that the IS—LM equilib-

rium part of the model determines the price level p and the nominal rate ofinterest r, again for both the Keynes—Wicksell model and the Keynesianmodel, in the same way. The IS—LM equations are thus solved for the

� Note here that the implied equality of investment I and total savings S allows theparameters �

�, �

�of the disequilibrium version of the Keynes—Wicksell model to be

ignored.


variables p and r on the basis of the given potential output of firmsY�� y�K. This is a very Friedmanian solution of the conditions for goods-and money-market equilibrium (though not one with full employment ofthe labor force). Perfectly flexible prices and interest rates here alwaysadjust aggregate demand to a predetermined level of aggregate supply, sothat quite obviously the model is no longer one of the Keynesian variety.��This strictly neoclassical structure of the fixed proportions case with

perfectly flexible prices is also present in the smooth factor substitutioncase, the only difference being that supply is now responsive to changes inthe price level and is determined in conjunction with goods and moneydemand on the basis of the equality of prices with marginal wage cost (inthe Keynesian story).��We conclude that perfect price flexibility is not compatible with a

Keynesian structure in models of monetary growth, but leads to a more orless obvious supply-side theory of the temporary equilibrium position ofthe economy, despite the formal presence of a Keynesian IS—LM block.The bastard case �

�� therefore inherits more from supply-sideKeynes-

ianism (the Keynes—Wicksell prototype model) than from demand-sideKeynesianism (our Keynesian prototype model). Though Keynes—Wick-sell models of monetary growth are allegedly considered as more or lessoutdated now (see, for example, the survey by Orphanides and Solow1990), they have therefore, in fact, implicitly survived in the literature in thewidely accepted form of so-called models of Keynesian dynamics, bysuppressing their questionable treatment of goods-market disequilibriumthrough infinitely flexible prices�� and by ignoring at the same time theirfundamental supply-side orientation. Simply assuming money-wage rigid-ity, as in the Keynesian variant of the neoclassical synthesis (as is generallydone, see Sargent 1987, chs. 1—2, for example), is insufficient to arrive at atruly Keynesian prototype of monetary growth. Such a prototype is in factmissing in the literature on monetary growth dynamics (see again thesurvey by Orphanides and Solow 1990).In order to solve�� the Keynesian variant of the neoclassical synthesis

�� This adjustment process is but a refinement of the Classical view on the role of the rate ofinterest to always equate aggregate savings with aggregate investment at the full capacitylevel (see Flaschel 1993, ch. 2, for details). This version of Say’s Law is explicitly criticized inKeynes (1936, chs. 2 and 14) as the main fault in the Classical theory of employment andoutput.

�� And through the equality of the real wage with the marginal product of labor in theKeynes—Wicksell case — which, however, does not make a difference to the Keynesian casein the formal investigation of the model.

�� By assuming goods-market equilibrium throughout, the Wicksellian theory of inflation isonly present in the background of the model and, if at all, only considered explicitly as anultra short-run adjustment mechanism, as in Sargent (1987, ch. 2).

�� We set �� 0(g� t�) as in Sargent (1987) for simplicity.


presented above, one has first of all to choose the appropriate state vari-ables for the intensive form of the model. Here, it is to be noticed that thevariable m�M/(pK) can no longer serve this purpose, since it has nowbecome statically endogenous. As state variables it is now appropriate tomake use of l, �, and v�M/(wK), i.e., to use now wage units in themeasurement of real balances per unit of capital. The IS—LM, or better theAD—AS, equations then allow the determination of real balances m, thenominal rate of interest r, the real wage �, the labor intensity l, the rate ofprofit �, and the output—capital ratio y from the temporary equilibriumpart of the model as functions of the dynamically endogenous variables vand �. On the basis of these functional dependencies, one thereby gets inthis case of perfectly flexible prices the autonomous dynamical system

l1 � n� s�(�� t�), (5.67)

v�� n� �� l1 ��

�(l/l� v� ), (5.68)

�� 1

1�m�/m��[��(�

� n�� l1 �m�/mv� )]. (5.69)

This, now explicit, system of differential equations can be treated as theoriginal Sargent (1987, ch. 5) model was treated in Franke (1992a), and willgive rise to the same Hopf-bifurcation situation obtained by the latterauthor if the parameter �� is chosen sufficiently large.Sargent (1987, ch. 5) also considers a further limit case of this model

where �� holds (i.e. � � p, the case of myopic perfect foresight). In thiscase we lose the third dynamical law of the system (5.67)—(5.69) and getfrom the money-wage Phillips curve of the model a real-wage Phillipscurve � ��

�(l/l�V� ). This new representation of the Phillips curve pro-

vides the reason why Sargent (1987, ch. 5) now considers the real wage �,and no longer themoney wage, as a new state variable of the system. Hencewe now can employ this variable � in the place of the above v in thedescription of the dynamics of this limit case of the above model so that weobtain

l1 � n� s�(�(�)� t�, (5.70)

�� (l(�)/l�V� ). (5.71)

Note here that both the rate of profit and the employed labor intensity aredecreasing functions of only the real wage now (because of the marginalproductivity rule and the assumed neoclassical production function). Thisimplies that the dynamical model (5.70)—(5.71) is self-contained. It is in factnow a simple Solovian real growth model combined with a sluggishadjustment of real wages as in the Goodwin growth cycle model. This


simple dynamical model implies a globally asymptotically stable adjust-ment process towards the Solovian steady state. In this connection seeFlaschel (1993, chs. 4, 6, and 8) who shows in particular that the model isnow a completely supply-side determined one, since the Keynesian IS—LMsector does not appear at all in its laws of motion.��Incorporating perfect price level flexibility into the Keynesian model of

chapter 4 raises the question of why this is not also admissible for wagelevel flexibility. Indeed, wage levels and price levels should not be treated inthis strictly asymmetric way. Instead, if one allows for the possibility of aperfectly flexible price level (which instantaneously adjusts and therebyremoves any imbalance in the use of the capital stock or on the market forgoods), one should also be prepared to allow for complete wage flexibility.Alternatively one may accept that both the wage level and the price levelactually need at least some time to adjust to imbalances in the market forlabor and for goods (as we have assumed in our Keynesian prototypemodel).Allowing for this further limit case, now in the adjustment of wages, gives

rise to the law of motion

l1 � n� s�(�(�(l))� t�),

as an extremely simple supply side response of the model. This is so sincethe real wage can nowbe determined from the equilibrium condition on themarket for labor l� l(�)/V� as a function of the full employment laborintensity l.Instead of proceeding in this way tomore andmore extreme joint special

cases of the Keynes—Wicksell as well as theKeynesian prototypemodel, wewould, however, maintain that the Keynesian model is much closer toreality if the alternative direction is taken, namely, that of a price as well asa wage level that are at least somewhat sluggish in their reaction toimbalances in the utilization of the capital stock and the labor force,respectively. We therefore propose to leave the above border situationsbetween supply-sideKeynes—Wicksell and demand-sideKeynesianmonet-ary growth models and to return to the adjustment equations for wagesand prices we have used throughout this book, namely,

w� ��( · )�

�p� (1�

�)��

��

�( · )�

�(p��

�),

p� ��( · )�

�w� (1�

�)��

��

�( · )�

�(w��

�).

�� It is used in this model for the sole purpose of determining the rate of inflation p, (and hencealso the rate of change of real balances m) via IS—LM equilibrium and thus through anadjustment of aggregate demand toward predetermined aggregate supply by appropriatechanges in this rate of inflation and the nominal rate of interest. TheWicksellian characterof this procedure is again obvious.


Note that we have here allowed for separate expressions for expectedmedium-run price and wage inflation �

�and �

�. These adjustment equa-

tions yield a complete description of the wage—pricemodule by assuming inaddition and in a symmetric fashion the two mechanisms for changes inwage and price expectations given by

��[��w� (1� �

�)(

�� n)—�

�], �

�� [0, 1],

��[��p� (1� �

�)(

�� n)—�

�], �

�� [0, 1].

The interpretation of these four dynamical laws is the same as before, interms of a mixture of forward- and backward-looking behavior of nowboth wage earners and firms. Besides the assumptions �� , ��, we havethen to assume only that

�and

�are not equal to 1 at one and the same

time (which excludes that both prices and wages exercise a full cost-pushimpact effect on each other)� and that �

�and �

�are not equal to one at 1

and the same time (which excludes that both wage earners and firms arepurely backward looking in their behavior). We may even allow adjust-ment of expectations under these circumstances to be infinitely fast(�� , �� ) without obtaining supply-side models as we have consideredthem above.We observe in closing this section that nearly all textbook treatments of

wage—price dynamics (the medium run) and growth dynamics (the longrun) are not obtained as a proper extension of their short-run model,generally the Keynesian IS—LM model, but are in fact in contradiction totheir theory of the short run, by removing part or all of the determinationof income and nominal interest of the IS—LM approach from their theoryof the medium and the long run.

5.4 Outlook: sluggish price as well as quantity dynamics

Assuming smooth factor substitution as in the present chapter has in-creased the descriptive contents of the Keynesian model of monetarygrowth, yet has not essentiallymodified its theoretical structure, as we havejust seen. In appendix 2 of chapter 4, we have already analyzed theextensions of this prototype model which add technology change, morerefined expectations mechanisms, and wage taxation in particular, butwhich assume the less complex case of a fixed proportions technology. Wesaw that these changes add descriptive realism to the Keynesian model ofmonetary growth, without changing its structure in an essential way.

� Otherwise, capacity utilization levels on the labor market and within firms move inverselyto each other, or there would be a contradiction between thewage and the price adjustmentequation.


We have considered in chapter 4 and in this chapter various extensionsof the Keynesian prototype model of monetary growth that have added toits descriptive content (or its realism when used as a framework for macro-econometricmodel building). Each of these extensions was important in itsown right, but not of decisive importance when the theoretical soundnessof the chosen approach toKeynesianmonetary growth is themain issue. Inthis respect the following chapter will, however, be of decisive importance,since it will remove one final basic defect (one problematic asymmetry)from our Keynesian prototype model that is responsible for some peculiarproperties of the basic prototype which we observed in chapter 4. Inparticular, the three different equilibrium subcases of this approach and theinstability phenomena surrounding two of them.These subcases, in fact, cease to be of any importance (they simply do not

occur) when another important descriptive element of Keynesian dynamicsis taken into account. Besides a symmetric treatment of labor and capitalutilization, we should also allow for a symmetric treatment of price andoutput adjustments, by allowing for (somewhat) sluggish output adjust-ment besides the sluggish price and wage adjustment so far assumed. Thisnew element not only adds further realism to our Keynesian model ofmonetary growth (since we allow thereby for the fact that firms do notalways have perfect sales expectations), but it also leads to a dynamicanalysis of this revised model which is significantly different from that ofchapter 4, and which is not subject to any subdivision into equilibriumregimes and their instability consequences discussed there.There are two ways in which a sluggish output adjustment can be

introduced into our IS—LM-equilibrium model of monetary growth. Thefirst is along the lines of Kaldor’s (1940) analysis of the trade cycle andTobin’s (1975) analysis of medium-run price dynamics, namely, by usingthe conventional dynamicmultiplier in the place of the static one of IS—LMequilibrium.The second is a proper integration of theMetzlerian inventoryadjustment mechanism into our Keynesian model of monetary growth.From a mathematical point of view, the first approach is generally prefer-able to the economically preferable alternative approach. However, thesecond approach is the only one that can be considered as internallyconsistent. Nevertheless, conditions can be provided where it reduces tothe first approach, which is then easier to analyze due to the fact that itsdynamics are of a lower dimension.With this revision of the Keynesian prototype model of monetary

growth, we therefore finally arrive at the model type which can be consider-ed as the workingmodel of theKeynesian analysis of monetary growth andbusiness fluctuations. Of course, further modifications and extensions ofthis model type will also be needed in the future to make some or all of the


modules of the model of amoremodern type.�Yet none of thesemodifica-tions will require the dismantling or recasting of what we will have estab-lished by the end of the next chapter. Namely, the basic framework for aKeynesian analysis of monetary growth of both theoretical as well asempirical relevance.Further possibilities for extensions and refinements of the chosen frame-

work will be discussed in the final chapter of this book. There we shall alsoprovide further examples of how the model type of chapter 6 can beextended still further in the direction of more descriptive content withoutlosing sight of the theoretical framework of Keynesian monetary growthdynamics that has been systematically developed in this book. Takentogether, chapters 5, 6, and 7 therefore provide modifications of the basicKeynesian prototype model of monetary growth (two secondary ones andone that is essential) that add realism to it, and that should be integratedinto an overall approach to Keynesian monetary growth dynamics whenthis model comes to be used for macroeconometric purposes. In thistheoretically oriented book, such an integration (which, in fact, is easy toperform) is not, however, provided, in order to keep the presentation of thevarious model types on a less involved and technical level.

� For example, the simple type of cost-push and demand-pull wage and price inflation wehave considered so far surely needs further discussion.


6 Keynesian monetary growth: theworking model

6.1 Introduction

In this chapter we continue to build the Keynesian prototype model ofchapter 4 on a firmer and more general basis. We will find that the moregeneral model of this chapter, at one and the same time, avoids theseparation into three types of comparative static analysis of the IS—LMequilibrium part of the model (section 4.2) and also avoids the ultrashort-run stability problems observed in section 4.3.We have already extended the Keynesian prototype model of chapter 4,

for wage taxation, technical change, average inflation, and a more refinedconcept of forward-looking behavior, the so-called p*-concept (see appen-dix 2 of chapter 4), and in chapter 5 by allowing for smooth factorsubstitution. Although these extensions are all important from an empiri-cal point of view, we consider them as secondary as far as the conceptualissue of building a proper Keynesian model of monetary growth is con-cerned. This is also obvious from the fact that these extensions can beapplied to all three of the general prototype models of this book (chapters2—4) in a uniform way.In the present chapter we now show how the simple goods-market

disequilibriumapproach of the Keynes—Wicksellmodel of chapter 3 can beintegrated into the Keynesian, or IS—LM, model of monetary growth, stillbased on goods-market equilibrium, in chapter 4. In this way we now willarrive at a general prototype or working model of Keynesian monetarygrowth that is based on sluggish price/wage as well as sluggish quantityadjustment processes, and which thereby avoids the awkward ultra short-run problems (and their comparative static counterpart) of the one-sideddisequilibrium adjustment processes of the Keynesian prototype model ofchapter 4.In sum, the model of this chapter can therefore be considered as the

synthesis of the Keynesian models of chapters 3 and 4, which attempted to

278

provide an alternative to the neoclassical model of monetary growth thatwe provided in a general format in chapter 2. This new monetary growthmodel is of Keynes—Metzler type and (as its name implies) in particularadds aMetzlerian inventory adjustment mechanism (based on sales expec-tations) to the Keynesian model of chapter 4. It treats goods-marketdisequilibrium as causing a set of quantity adjustment processes and this,in contrast to the Keynes—Wicksell model type, for less than full capacitygrowth in general. Price and wage inflation is again driven by the levels ofcapacity utilization on the market both for goods as well as for labor.However goods-market (IS) disequilibrium proper initially only causesinventory considerations by firms which, only in a second phase andcoupled with sales expectations, are transmitted to the use of productivecapacity and to the dynamics of the price level.Of course, the extensions of chapters 4 and 5 should be added also to the

present Keynes—Metzler extension of the Keynesian model. The incorpor-ation of such extensions, however, can be achieved in an obvious way, as inchapters 4 and 5, and will not be repeated here in order not to overload thestructure of our working Keynesian model. Finally, we will discuss in thenext chapter a list of the shortcomings and omissions that are still con-tained in the working model type developed in this chapter. Thus theworking Keynesian model developed in this chapter must still be consider-ed as an intermediate stage in the development of a macrodynamic modelthat can be considered as satisfactory in all respects.As will be obvious from the list of shortcomings presented in the final

chapter 7 (see section 7.7), it is not possible to remove all of these from ourworkingmodel in one single book.We will thus concentrate in this chapterbasically on one of its crucial problematic characteristics, namely, theasymmetry that has existed so far in the treatment of the speed of adjust-ments of prices and quantities.We have assumed that quantities are alwaysmarket clearing in the sense of the IS—LM equilibrium theory of effectivedemand and thus always adjust with infinite speed, while prices and wagesadjust more or less sluggishly, responding to the disequilibria that exist inthe employment of the services of capital and labor.In chapter 4 we based price level changes on the imbalance caused by

goods-market (IS) equilibrium within firms between their actual and theirdesired rate of capacity utilization. This was an improvement on theproblematic I�S disequilibriumapproach to the theory of inflation adop-ted by Keynes—Wicksell models of chapter 3. We shall allow again forIS-disequilibrium as in the Keynes—Wicksellmodel type; now, however, byadding firstly a Kaldor (1940) quantity dynamics and, secondly, a Metzler(1941) quantity adjustment process to themodel. This corrects and general-izes at one and the same time the Keynes—Wicksell and the Keynesian

279The working model

prototype model by an appropriate synthesis of the two model types andtheir theory of price inflation.We shall show in chapter 7, finally, how, for the Kaldorian adjustment

process, further important rigidities that characterize all of ourmodel typesso far can be removed. These final improvements concern the exogeneity ofthe full rate of employment V� and the natural rate of growth of theeconomy which we have assumed so far as given. We shall refine theadjustment processes on the labor market and endogenize the trend com-ponent in the investment function and on the labor market in a way thatallows for an endogenous determination of the steady-state rate of growthas well as the steady-state rate of (un)employment.One consequence of our choice of the endogeneity of these rates will be

that there is then hysteresis in the dynamics, i.e., these rates are no longeruniquely determined and there is then also path dependence in the long-run behavior of the trajectories of the dynamics. Also, in order to allow foran adjustment of the NAIRU rate of employment, we have to distinguishthen, on the one hand, between the rate of employment that refers to thelabor market and, on the other hand, the one within firms (the rate ofemployment of the employed labor force). This introduces further delays inthe adjustment of the rate of employment on the labor market (by distin-guishing inside from outside effects), and it makes the description ofemployment relationships also more realistic.In sum, we will therefore in chapter 7 eliminate the use of so-called

natural rates in the formulation of our Keynesian monetary growth dy-namics by introducing appropriate adjustment processes for them. Inaddition, we will also base the model on finite adjustment speeds caused bygoods-market disequilibrium (introduced in the present chapter), thoughwe here choose a particularly simple version of it. All othermarkets (i.e., theasset markets) are still assumed to be in equilibrium in this final chapter ofthe book, in exactly the same way that they have been modeled in theprevious chapters.Before turning in the present chapter to a full description of goods-

market disequilibrium and its dynamic consequences, we consider the‘‘naive’’ Keynesian version of the goods-market disequilibriumadjustment,i.e., the simple dynamic multiplier story of output adjustments. This ver-sion considers goods-market disequilibrium and its consequences as in thewell-known Kaldor (1940) trade cycle model or in the model of a Key-nesian depression of Tobin (1975) and will be called the Kaldor—Tobin(KT) model for these reasons.�

� This model represents the simplest case of a monetary growth model with sluggish wage,price, as well as quantity adjustments that allows the discussion of price/output stabilityissues as in Tobin (1975); see also Tobin (1992, 1993) on this matter.


This version of the Keynesian model with both sluggish price andquantity adjustments is mathematically simpler to treat and, though un-convincing from the viewpoint of economic consistency, not unrelated tothe mathematical structure of the monetary growth model with a fulldescription of goods-market adjustment processes along Metzlerian linesthat follows later on. Indeed, we shall see that a simple reformulation of thefuller model will give rise to a dynamic structure that is very similar to theone with the dynamic multiplier story. This latter and simpler approach,though economically unconvincing, may thus nevertheless reveal import-ant dynamic properties in a setup that contains only a five-dimensionaldynamical system rather than a six-dimensional one, as in the model withfully elaborated quantity adjustment processes.�Due to the high dimensions of the dynamical systems considered in this

chapter, there will only be few analytical results possible here. Of coursethere always remains the possibility of investigating typical two- or three-dimensional isolated subdynamics of such general approaches by disen-tangling them via appropriate assumptions, as we have done in earlierchapters. General results are here concentrated on the fact that suchKeynesian models of monetary growth allow for a great variety of situ-ations where the Hopf bifurcation theorem can be applied, based on thegenerally given fact that the determinant of the Jacobian (evaluated at thesteady state) of each of the considered systems of chapters 2—7 is nonzeroand of the sign demanded by the Routh—Hurwitz conditions for localasymptotic stability.We will show here that the types of Hopf bifurcationthat occur in the general dynamics will not have much in common withthose that characterize the isolated subdynamics. From a mathematicalpoint of view this is not very astonishing, due to the difference that existsbetween the full dynamics and the isolated subdynamics. But from aneconomic perspective this implies that the pure integration of known two-or three-dimensional prototype (real, monetary, or inventory) dynamicalprocesses will lead to a dynamic behavior of the integrated dynamics thatcannot be judged from these lower-dimensional prototypic, but partial,subdynamics. This raises the question of the meaningfulness of consideringreal or monetary or inventory adjustment processes in isolation.Finally, it comes as not unexpected that the five- or six-dimensional

dynamical models that we will investigate in this chapter are capable of

� In the same way, it may be shown that the two-dimensional Kaldorian trade cycle modelmay be very similar in its implications to the three-dimensional Metzlerian extension of itwhich considers the two state variables ‘‘sales expectations’’ and ‘‘inventories’’ in the place ofonly one: ‘‘output’’ (which again puts such disequilibrium analysis on a more convincingbasis, but which also makes it dynamically more complex).

In the dynamic models of chapter 7 this only holds for the subdynamics where hysteresiseffects (which imply a zero overall determinant) have been removed.


producing ‘‘chaotic’’ dynamics, in particular since they combine cyclemechanisms in the real, the monetary, and the inventory subsectors thatare fairly independent of each other. From an economic point of view,however, this seems to occur only in situations of a more or less extremenature. For example, in a state space domain where economic viability ofthe considered dynamics is already violated and which seems to be at theborder of mathematical viability as well. Alternatively, chaotic dynamicscan occur within the domain of economic viability by making use ofadjustment speeds of prices, wages, or expectations that seem very large,and by adding appropriate bounds to these price dynamics.This latter observation needs further explanation. As in chapters 2—5, we

at first consider our monetary growth dynamics with only linear behav-ioral relationships, i.e., we only allow for natural or unavoidable non-linearities as they come about through growth rate formulations or prod-ucts or quotients of certain state variables caused by value expressions,typical ratios of growth theory, and the like. In these only intrinsicallynonlinear situations, a conclusion based on many simulation runs of thesemodels is that limit cycles or superimposed limit cycles are the most thatone can generally expect to obtain for the dynamics under consideration.Only at the edge of mathematical viability of the observed attractors doesone find more complex dynamical patterns, as we shall demonstrate lateron in this chapter.The need arises to add further nonlinearities, perhaps motivated by

nonlinearities in the behavior of economic agents, to the considered dy-namics in order to make them viable from an economic point of view andalso with respect to broader parameter ranges of themodels.We have donethis in chapters 2—5 from various perspectives concerning investmentbehavior, price and wage adjustment speeds, and nonlinearities in thespecified technological relationships. In this chapter we now introduce afurther and very basic nonlinearity, namely a simple kinked Phillips curvethat takes account of the stylized fact that the growth rate of the level ofnominal wages can easily become positive, but only rarely becomes nega-tive. This very basic institutionally determined nonlinearity in the behaviorof wage levels will allow us at one and the same time to increase dramati-cally the domain of viability of the considered dynamics as well as to showhow chaotic attractors will come about in such situations.Nevertheless, as will be shown, a period-doubling route to chaos will

also then only come about when, for example, the adjustment speed ofnominal wages (when growing) becomes very high. Therefore, sufficientlysluggish wages (and prices) generally only create a fairly ‘‘ordinary’’ dy-namics of persistent fluctuations, despite the presence of various interac-ting cycle mechanisms. We conclude from this that integrated Keynesian


models of monetary growth (of dimension 4 to 8) exhibit dynamic feedbackchains that still interact with each other in very simple loops. Nevertheless,this integration produces dynamical features that do not mirror the dy-namical features of its partial components models.The next section introduces the Kaldor—Tobin model of monetary

growth and briefly discusses its stability properties. Section 6.3 then ex-tends this model to includeMetzlerian inventory adjustments, and thus wearrive at our working model of Keynesian monetary growth. This sectionalso provides stability considerations that relate appropriate subdynamicswith the full six-dimensional dynamics of this model type. Section 6.4finally considers a monetary growthmodel that represents an intermediatecase between the models of sections 6.2 and 6.3, and in particular showsthat this dynamic model can give rise to interesting numerical simulationswhen a very basic extrinsic nonlinearity is added to it. We conclude thischapter with some observations on the relationship of themodels discussedto macroeconometric model building.

6.2 The Kaldor–Tobin model of monetary growth

In this section we consider a simple possibility of extending the IS—LMequilibrium framework of chapter 4 to a treatment of goods-market dis-equilibrium of a Keynesian type which makes use of sluggish quantity aswell as price adjustment processes in the market for goods. This is incontrast to the conventional purely nominal Keynes—Wicksell treatment ofthe consequences of IS-disequilibrium situations that we considered inchapter 3. This section will therefore provide improvements of the generalmodels of both chapters 3 and 4 by introducing a Keynesian dynamicmultiplier treatment of goods-market adjustment processes. Moreover, itrepresents an important intermediate step in the derivation of an elaborateand fully consistent treatment of goods-market disequilibrium — see theKeynes—Metzler model in the next section — that helps in the understand-ing of the structure and implications of this core model of our book (seealso its alternative formulation in the section 6.4).Prominent examples of the dynamic multiplier approach to the treat-

ment of goods-market disequilibrium as a component part of a largerdynamical system are the real two-dimensional trade cycle model of Kal-dor (1940) and the medium-run three-dimensional dynamical model ofKeynesian depressions and recessions of Tobin (1975). These two modelshave both been extensively discussed in the literature on Keynesian dy-namics. These two approaches to macroeconomic dynamics are here atleast partially embedded into a full Keynesian model of monetary growth,and thus are allowed to interact with each other to some extent. We stress,


however, that the stationary Kaldorian approach to business cycle theoryexhibits some problems when put into the context of a growing economy(see in particular Skott 1991 on this matter). We also stress that this modeltype represents only an intermediate step to a complete and consistentKeynesian model of monetary growth with sluggish price as well as quan-tity adjustments. Due to its origins we will refer to this model type as theKaldor—Tobin (KT) model in the following discussion.

The equations of this model of monetary growth are:


��w/p, u�w/x,� � (Y� �K��L)/K, (6.1)


%� 1. (6.2)



�pY� h

�pK(1� �)(r� � r), (6.3)

C� �L� (1� s�)[�K� rB/p�T ], s

�� 0, (6.4)

S��L�Y�


� s�[�K� rB/p�T ]� s

�Y��

� (M� �B� � pE� )/p, (6.5)

L1 � n� const. (6.6)


Y�� y�K, y�� const.,U�Y/Y�� y/y�, (y�Y/K), (6.7)


I� i�(�� (r� �))K� i

�(U�U� )K� K, � n, (6.9)

pE� /p� I� (S� I)�Y� �K�C�G�Y�Y� I, (6.10)

K1 � I/K. (6.11)


T� �(�K� rB/p), [or t�� (T� rB/p)/K� const.], (6.12)

G�T� rB/p��M/p, (6.13)


M1 ��, (6.15)

The parameter � has to be removed from all equations of the following model if the secondalternative in equation (6.12) is chosen as the tax collection rule.


B� � pG� rB� pT�M� [� (��

�)M]. (6.16)



�pK(1� �)(r� � r)[B�B,E�E], (6.17)

pE� (1� �)�pK/((1� �)r��), (6.18)

M� �M� ,B� �B� [E� �E� ]. (6.19)

6 Disequilibrium situation (goods-market adjustment):

S� pE� � S

��S

��Y� �K�C�G� p

E� � I, (6.20)

Y�C� I� �K�G, (6.21)

Y1 � ��(Y/Y� 1)� ��

�((I� S)/Y), � n, (6.22)

N� � ��K� S� I,S�S

�� S

��Y� �K�C�G,

N inventories. (6.23)


w� ��(V�V� )�

�p� (1�

�)�, (6.24)

p��(U�U� )�

�w� (1�

�)�, (6.25)

�� (p��)��(�� n� �). (6.26)

This extendedmonetary growthmodel of the Keynesian type reverts to theinclusion of goods-market disequilibrium as it was present in the Keynes—Wicksell model, but not in our Keynesian reformulation of it in chapter 4.Equations (6.10) and (6.20) introduce such a disequilibrium in quantitiesdemanded and supplied on the market for goods, which is made compat-ible with asset-markets equilibrium and motivated and interpreted in thesame way as the corresponding equations of the Keynes—Wicksell case inchapter 3. As is generally customary in the IS—LM literature (but not inapproaches of Sargent 1987, Barro 1994a, and others toKeynesian disequi-librium analysis) the present IS-disequilibrium approach is now related toquantity adjustments in the production of firms, and not, as in the Keynes—Wicksell model (and in the related models of Barro and Sargent), to pricelevel adjustments. These latter adjustments are based here, as in the basicKeynesian prototype of chapter 4, on capacity utilization levels in theirdeviation from normal capacity utilization. The degree of capacity utiliz-ation U itself is changed in the light of the excesses or shortages on theoutput market.In order to provide a description of the adjustment process of U in the

light of observed goods-market disequilibrium, equation (6.21) first of alldefines the level of aggregate demand by the sum of its various compo-


nents. Note here that the following equivalent representation of excessdemand holds: Y�Y� S� I which provides a bridge to the equation(6.20) that describes the type of IS-disequilibrium that prevails in thepresent version of themodel. The next equation (6.22) gives a description ofthe simple dynamicmultiplier process in the context of a growing economywhere the expansion or contraction of production (its growth rate) isgoverned by two principles. These are goods-market imbalances and anincorporation of the rate of trend growth, which provides a very simpleintegration of the effects of persistent growth into this quantity adjustmentprocess. The final new equation in this model in comparison to its prede-cessor model in chapter 4 is given by equation (6.23), which, however,represents but a return to the simplistic (purely appended) treatment ofinventories of the Keynes—Wicksell model of chapter 3. Yet, such anapproach is typical for dynamic multiplier analyses, which often do noteven mention the inventory changes that are implied by their quantityadjustment process.�Note here, finally, that we have returned to technolo-gies with fixed proportions in this chapter.This concludes the simple inclusion of quantity adjustment processes

into our Keynesian prototype of monetary growth. The main aim ofincluding this section in the present chapter lies in its comparison informulation as well as implications with the much more consistentlyformulated Metzlerian inventory adjustment process for growing econo-mies which we shall construct in section 6.3.�In the the general case of an endogenous determination of taxes per unit

of capital, and thus of the existence of a feedback mechanism of govern-ment debt accumulation B(t) on the rest of the system, we obtain fromcalculations similar to those in the case of the basic Keynesian prototype(4.1)—(4.23) the following autonomous, now six-dimensional, dynamicalsystem in the variables ��w/p, l�L/K,m�M/(pK), �, b�B/(pK), andy�Y/K:

�� [(1� �)��(V�V� )� (

�� 1)�

�(U�U� )], (6.27)

l1 � n� i( · )�� i�(� � r��)� i

�(U�U� ), (6.28)

� Blanchard (1981, pp.132—133) offers two interpretations for this dynamicmultiplier rule, butthen observes: ‘‘A more satisfactory, and more complex formulation would allow forinventories to be rebuilt later during the adjustment process.’’

� Note here with respect to this intermediate case between goods-market equilibrium and aMetzlerian treatment of goods-market disequilibrium that this model version still assumesan income concept that is based on supply and not on (expected) demand, and that theassumption �

�� will lead us back to the Keynesian prototype model of chapter 4, thus

providing a first test of whether the assumption of goods-market equilibrium of the basicKeynesian version is really justified.

Note that the law of motion for the dynamic variable z, the stock of inventories, is againsimply ignored in this version of IS-disequilibrium dynamics.


m� �� n� [�

�(U�U� )�

��(V�V� )]� l1 , (6.29)

�� [��(U�U� )� ��(V�V� )]��(�

� n��), (6.30)

y� � l1 y� ��(y� y), (6.31)

b� � (��

�)m� (�� n)b� [ (�

�(U�U� )

� ��w(V�V� ))� l1]b, (6.32)

where we employ again the standard abbreviations (r� � r�):

y��y/x� (1� s�)(�� t�)� i

�(�� r� �)� i

�(U�U� )

� n� �� g,V� l/l,U� y/y�, l�L/K� y/x (y not const.!),�� y� ��l� y(1��/x)� �,r� r

�� (h

�y�m)/(h

�(1� �)),

t�T/K� �(� � rb), t�� t� rb,g� t��

�m,

s( · )� s�(�� t�)� (g� t�)�K1 � I/K� i( · )� n.

Let us now again assume tn � t � rb � const. and remove the parameter �from the equations of the model (since taxes are now lump sum). Further-more, we set U� �V� � 1 for notational simplicity. This gives a five-dimen-sional dynamical system in the variables��w/p, l�L/K,m�M/(pK), �,and y�Y/K—with an appended b� dynamical system, since the influence ofb on s( · ) and g (and thus on y) is now suppressed (y and U, V, �, r, asgiven above):

�� [(1� �)��(V� 1)� (

�� 1)�

�(U� 1)], (6.33)

l1 �� i�(�� r��)� i

�(U� 1), (6.34)

m� �� n� [�

�(U� 1)�

��(V� 1)]� l1 , (6.35)

�� [��(U� 1)� ��(V� 1)]��(�

� n��), (6.36)

y� � l1 y� ��(y� y). (6.37)

This is a Keynesian dynamic multiplier system which exhibits a delayedquantity adjustment besides sluggish price adjustment. We shall brieflyinvestigate this model from the analytical point of view in the remainder ofthis section before we turn to the Metzlerian extension of it. From aKaldorian perspective, one would have to investigate the isolated y, ldynamics in order to see whether the Poincare—Bendixson theorem canagain be applied to show the existence of persistent cyclicalmotions for thissubdynamics. Tobin (1975), furthermore, investigated the m, �, y sub-dynamics in order to show the existence of corridor stability in particular.This leaves as complement (or as a bridge between these two prominent


models of Keynesian dynamics) the �, l subdynamics of the Rose employ-ment cycle model. The above dynamic model can thus be subdivided in anumber of ways to give rise to prominent, though always partial, models ofmacrodynamics of the literature on Keynesian dynamics.The unique steady-state solution or point of rest of the dynamical system

(6.33)—(6.37) fulfilling ��, l�,m� 0 is given by:

y�� y

�� y�, l

�� l

�� y/x (6.38)

m�� h

�y�, (6.39)

��

�� n, (6.40)

��y�� t�� (n�

�m

�)/s�

l�

, (6.41)

�� y

��

�l�, (6.42)

r��

��

�� n. (6.43)

As in the preceding chapter, we assume that the parameters of the modelare chosen such that the steady state values for �, l, and m are all positive.

Proposition 6.1: The determinant of the Jacobian J of the dynami-cal system (6.33)—(6.37), evaluated at the steady state, is always negative.

Proof: Since linear combinations of rows of J can be added to itsrows without changing det J, the system (6.33)—(6.37) can be reduced to thefollowing form without change in the determinant of the system:

�� (1� �)��(V� 1),

l1 �� i�(�� r),

m� � ��(U� 1),

�� (�� n��),

y� ��(y& � y),

where y& � �y/x� (1� s�)(�� t�)� n� � � g, and V, �, U, and r are as

described above. Proceeding along these lines one then gets (sincey& � y�� s

�� g� const.)

det J� �0 � 0 0 �

� 0 � 0 ?

0 0 0 0 �

0 0 0 � 0

� 0 � 0 ��

0 � 0 0 0

� 0 � 0 0

0 0 0 0 �

0 0 0 � 0

� 0 � 0 0�


��

0 � 0 0 0

� 0 0 0 0

0 0 0 0 �

0 0 0 � 0

0 0 � 0 0��

� 0 0 0

0 0 0 �

0 0 � 0

0 � 0 0 �� 0 0 �

0 � 0

� 0 0 �� 0.

�

This proposition implies that the steady state of the dynamics (6.33)—6.37)will, except for a set of measure zero in the model’s parameter space, onlylose its stability by way of a Hopf bifurcation, since no eigenvalue can passfrom the left part of the complex plane to its right part by going through theorigin.

Proposition 6.2: (1) The real sector�, l subdynamics of the dynami-cal system (6.33)—(6.37), with m, �, and y frozen at their steady state values,is of Goodwin (1967) growth cycle type; (2) The subdynamics m, �, y of thedynamical system (6.33)—(6.37), with � and l frozen at their steady statevalues, is of what Tobin (1975) calls WPK type, if in addition �� 0,�� 0,

�� 1(p� w) is assumed.

Proof: (1) The real sector subdynamics is represented by

�� (1� �)��(y

�/(xl)� 1),

l1 �� i�(y

�(1��/x)� �� r

��

�),

which is of the same cross-dual nature as the Goodwin (1967) growth cycledynamics.(2) the above Tobin-type subdynamics of (6.33)—(6.37) is given by

m� �� n� �

�(y/(xl

�)� 1),

�� (y/(xl

�)� 1),

y� ��(y� y),

where aggregate demand y is given by

y � ��y/x� (1� s

�) (� � t�)� i

�(� � r� �)� i

�(y/y� � 1)

� n� � � g

with �, r, and g as described above (� ��).

The Jacobian of this dynamical system evaluated at the steady statereads


J��0 m� � m�

�0 0 ��

�y��

y� � y�� ,

fromwhich we obtain for the Routh—Hurwitz coefficients a#of the Jacobian

at the steady state:

�a�det J� ��

�y��m� �� 0,

a��det J

�� det J

��det J

�� (��

�y� � �m�

�y��)� 0,

�a�� trace J� y

�� 0.

The latter inequality holds if it is assumed, as in Tobin (1975), that thedynamic multiplier considered on its own represents an asymptoticallystable process. Evaluating the expression for a

�further gives

a��

��[m

�y�� y�]/(xl�y�),

which provides an analog to Tobin’s (1975) critical condition for localasymptotic stability concerning the speed of adjustment of adaptivelyformed expectations, namely,

�� m�y�/y�.

This latter condition is again based on the Keynes effect y�� 0 and the

Mundell effect y�� 0, here simply with respect to aggregate demand in theplace of the effective demand concept of chapter 4.It is easy to show that the expression a

�a�� a

of the Tobin-type sub-

dynamics is a linear and strictly falling function h(��) of the parameter ��,which takes on a negative value at �� m

�y�/y� (where a�� 0 holds).

This subdynamics therefore undergoes a Hopf bifurcation (because ofa� 0) at a parameter value �� that is lower than the Tobin critical value��. Below ��, the system is always locally asymptotically stable, while this

is nowhere true above the value ��.�

In light of proposition 6.2, the full dynamical system (6.33)—(6.37) thereforecan be considered as representing a Goodwinian growth cycle that isinteracting with a Keynesian/monetarist dynamics as investigated in To-bin (1975) with respect to its local and global stability properties.We stress that the full five-dimensional dynamical system is not only

composed of the interaction of this three-dimensional Tobin and two-dimensional Goodwin subdynamics, but is also (if �

�� 0,

�� 1) deter-

mined by influences of the Rose (1967) employment cycle type, since it thenalso matters whether wages or prices are more flexible with respect tolabor- and goods-market disequilibrium (respectively). Furthermore, we


conjecture that the full dynamical system is locally asymptotically stablefor h

�, �� sufficiently small and �

�sufficiently large, if y�

�� 0 holds (i.e., if a

stable partial dynamic multiplier process is again assumed). If true, thisassertion generalizes a result we obtained in chapter 4 for IS—LM-equilib-rium to goods-market disequilibrium with its finite adjustment speed ofoutput. A specific form of this conjecture will be stated and proved below.We also expect that the five-dimensional dynamical behavior is generallydetermined by two independent cycle generating mechanisms: the Good-win—Rose (1967) growth/employment cycle and the Tobin (1975) Key-nesian/monetarist inflation/unemployment one.Another prominent cycle-generating mechanism, that of the Kaldorian

trade cycle, is, however, not present in the abovemonetary growth dynami-cs. Kaldor’s (1940) trade cycle model was based on the following twodynamical laws

Y� ��(I(Y,K)� �K� S),S� sY

K� � I(Y,K)� �K,

for output and capital stock adjustment, and it made critical use of aninvestment function I(Y,K) that was not homogeneous of degree 0 inY andK, but relied on a very special nonlinearity in such an investment function.Our dynamic approach is, however, based on such a homogeneity, whichin the case of Kaldor’s two dynamic laws would imply

y�Y1 �K1 � (Y� /K)/y�K1 ,��

�((I(y, 1)� � � sy)/y)� (I(y, 1)� �).

This is a single dynamic law in the output—capital ratio y which shows thatthere cannot be cyclical movements in this ratio y or its constituent partsYand K. The Kaldorian trade cycle mechanism thus is absent from thepresent form of a monetary growth dynamics, despite the presence of theKaldorian equation for goods-market adjustments.

Lemma 6.1: The three-dimensional subdynamics of the dynamicalsystem (6.33)—(6.37) that is obtained from it by freezing the real wage andinflationary expectations at their steady-state levels exhibits a locallyasymptotically stable steady state if the parameter h

�in the money demand

function is chosen sufficiently small and the parameter ��sufficiently large.

Proof: The considered subdynamics are given by the system ofdifferential equations

l1 �� i�(�� r��

�)� i

�(y/y�� 1), (6.44)

m�� [��(y/y�� 1)�

��(y/(xl)� 1)]� l1 , (6.45)


y� � l1 y� ��(y� y), (6.46)

where y is given by

y��y/x� (1� s

�)(y(1��

�/x)� �� t�)� n� �� g

� i�(y(1��

�/x)� �� r� �

�)� i

�(y/y�� 1).

Let us first calculate the sign of the determinant of the Jacobian J of thissystem at the steady state. To do this it suffices to consider the reduceddynamical system

l1 �� i�(�� r��

�)� i

�(y/y�� 1), (6.47)

m� � [��(y/y�� 1)�

��(y/(xl)� 1)], (6.48)

y� ��y/x� (1� s

�)(y(1��

�/x)� � � t�)� n� � � g� y.

(6.49)

This system can be further reduced, without change in the sign of therespective determinant, to the form

l1 �� i�m/h

�, (6.50)

m� � 1/(xl), (6.51)

y� �� s�y(1��

�/x). (6.52)

This last representation of the structure of the dynamics of the model(appropriately purified with respect to the many linear dependencies thatare contained in it) immediately implies a negative sign for the determinantof its Jacobian. In addition to the establishment of this Routh—Hurwitzcondition for local asymptotic stability it is also easy to show that trace Jmust always be negative.The further Routh—Hurwitz condition concerns the sign of the principal

minors of dimension two of the above Jacobian J. It is easy to show thatone of them must be positive and one zero, while the final one will becomepositive if the parameter h

�is decreased sufficiently, since the partial

derivative m�is made negative by such a choice and since y�

�is positive.

The remaining Routh—Hurwitz condition (a�a�� a

� (� trace

J)(J��J

�� J

)�det J� 0) is then easily shown to hold true as well if the

parameter ��is chosen sufficiently large, since all three components a

#of

this Routh—Hurwitz condition depend positively and linearly on this par-ameter value.�

Lemma 6.2: The four-dimensional subdynamics of the dynamicalsystem (6.33)—(6.37) that is obtained from it by freezing the real wage at itssteady-state levels exhibits a locally asymptotically stable steady state if the


parameter h�in the money demand function and the parameter �� are

chosen sufficiently small and if the parameter ��becomes sufficiently large.

Proof: Using the technique employed in the preceding lemma inorder to calculate the sign of the determinant of the Jacobian of thisdynamical system at the steady state, one can easily show that the sign ofthis determinant must be positive if �� 0 holds true and that it is zero ifthis parameter value is zero. In this latter case we have by the precedinglemma three eigenvalues with negative real parts and a fourth eigenvaluethat is zero. From continuity arguments, it then follows that this fourtheigenvalue must become negative when �� 0 is made positive (andchosen sufficiently small), due to the sign of det J.�

Proposition 6.3: The full five-dimensional dynamical system(6.33)—(6.37) exhibits a locally asymptotically stable steady state if theparameters h

�, ��, �

�, and �

�are all chosen sufficiently small, and the

parameter ��sufficiently large.

Proof:We have already shown in proposition 6.1 that det J mustbe negative for the full five-dimensional dynamics. The proof of proposi-tion 6.3 then follows from this fact by the arguments we have alreadyapplied in the proof of the preceding lemma.�

Note that a more detailed proof of proposition 6.3 is given in Chiarella andFlaschel (2000). It also follows from proposition 6.1 that the full five-dimensional dynamics will undergo a variety of Hopf bifurcations if theparameters h

�,�� and �

�or �

�are increased. This will be due to a

weakening of the stabilizing Keynes effect y�� 0 in the case of the par-

ameter h�, a strengthening of the destabilizingMundell effect y�� 0 in the

case of the parameter �� and is due to the Rose effect (which may bepositive or negative) in the case of the final two parameters. We stress thatthe parameter �

�does not seem to allow for a similar proposition in an

obvious way.We shall reconsider the above dynamical system in a slightly modified

form analytically and numerically in greater detail in section 6.4 after thetreatment of a proper respecification of the Kaldor—Tobin model as aKeynes—Metzler monetary growth model, where besides the dynamics ofsales expectations an inventory adjustment mechanism is added and re-places the above simple dynamic multiplier mechanism.

6.3 An integrated Keynes–Metzler model of monetary growth

When there is Keynesian goods-market disequilibrium as in the precedingsection there are disappointed sales expectations of firms, the revision of


which requires some explanation. Furthermore, there then necessarily existunintended inventories which lead to an intended adjustment of theseinventories that will generally deviate from actual inventory changes thatfirms will face. A consistent description of these basic ingredients of aMetzlerian inventory adjustment process will here be formulated in thecontext of Keynesianmonetary growth dynamics in order to investigate itsproperties in this setup.� We also wish to see to what extent such aconsistent formulation of quantity adjustments of firms can lead to devi-ations from the results obtained for the dynamic multiplier approachconsidered in the preceding subsection.

6.3.1 The model

The equations� of this Keynes—Metzler model of monetary growth are:��


��w/p, u��/x,�� (Y� �K��L)/K, (6.53)


%� 1. (6.54)



�pY� h

�pK(1� �)(r� � r), (6.55)

C� �L� (1� s�)[�K� rB/p�T ], s

�� 0, (6.56)

S��L�Y�

��C�Y� �K� rB/p�T�C� s

�[�K

� rB/p�T ]� s�Y��

� (M� �B� � pE� )/p, (6.57)

L1 � n� const. (6.58)




� See alsoChiarella andFlaschel (1998a) for a discussion of thismodel type,wheremore stressis laid on a consideration of the governmentbudget restraint and the occurrenceof complexdynamics.


�� See de la Grandville (1986), Blinder (1990), Franke and Lux (1993), and Franke (1996) forimportant discussions of theMetzlerian inventory mechanism in an IS—LM framework orin a Keynesian growth model. Our use of this mechanism in the following model is closelyrelated to Franke (1996). The role and extent of nonlinearities in this cycle mechanism isfurther discussed inMatsumoto (1995a,b), while the interaction of the inventory cycle withthe business cycle is investigated from the empirical point of view inFlood andLowe (1995).


I� i�(�� (r��))K� i

�(U�U� )K� K, � n, (6.61)

S��Y

��Y�Y�I, (6.62)

�Y�Y� �K�C� I�G�Y�Y, (6.63)

pE� /p� I� �Y� I� (N� �I), (6.64)

I�� I�N� � I� �Y�I� I��Y� pE� /p�I, (6.65)

K1 � I/K. (6.66)


T� �(�K� rB/p) [or t�� (T� rB/p)/K� const.], (6.67)

G�T� rB/p��M/p, (6.68)


M1 ��, (6.70)

B� � pG� rB� pT�M� [� (��

�)M]. (6.71)



�pK(1� �)(r� � r)[B�B,E�E], (6.72)

pE� (1� �)�pK/((1� �)r� �), (6.73)

M� �M� ,B� �B� [E� �E� ]. (6.74)

6 Disequilibrium situation (goods market adjustments):

S�Sp�S�� S

�� p

E� /p�I� I�N� � I�� p

E� /p�I,

(6.75)

Y�C� I� �K�G, (6.76)

N� ��Y,I� N��

�(N�N), � n, (6.77)

Y�Y�I, (6.78)

Y� � Y� ��(Y�Y), � n, (6.79)

N� �Y�Y�S� I[S� I��Y�Y]. (6.80)


w� ��(V�V� )�

�p� (1�

�)�, (6.81)

p��(U�U� )�

�w� (1�

�)�, (6.82)

�� (p��)��(�� n� �). (6.83)

The first thing we have to notice with respect to this revision of the dynamic


multiplier approach of the preceding section is that we now have todistinguish between production, demand, and expected demand on the onehand and between desired and actual inventory changes on the other. Inthe assumed disequilibrium situation on the market for goods (which wehave considered already in chapter 3 in the context of the Keynes—Wicksellmodel) we assumed that the income expectations of asset holders (as well asthose of workers) are based on actual production, though actual sales willgenerally differ from them. Firms were assumed to pay dividends based onthe output volume they produce since they did not have any demandexpectations different from it. Such an assumption may be appropriate inthe supply driven Keynes—Wicksell scenario, but it is surely inadequate forthe demand determined system we have considered in the preceding sec-tion. Yet, the dynamic multiplier story we have used in that section isexactly of this supply oriented type, where firms base their plans on currentproduction and only adjust this production (later on) in view of theaggregate they then face (which is based on the income and the profits thatflow from the side of production as if all output were always sold).In a demand determined systemwe should, however, base income expec-

tations on expected sales and not on actual output, which may followexpected sales with some lag due to the inventory adjustments that have totake place if expectations of sales are not fulfilled. Since workers get paidfor their actual work, which is based on actual production, income expecta-tions are only important in the case of asset-owning households. Here, the�-equation in (6.53) simply states that profit (dividend) expectations ofwealth owners are now based on expected sales and are always fulfilled (forthem), since firms are assumed to pay out these dividends independently oftheir actual sales. We thus get that only firms will suffer (or gain) frompossible errors in expectations, by so-called windfall losses or gains. Ofcourse, other assumptions on the distribution of the effects of unexpectedlyhigh or low demand are possible, but will not be considered here.Due to this assumption on the dividend payments of firms we get (as in

the goods-market disequilibriummodels of chapter 3 and as in the preced-ing section) the equation (6.64), again stating that the equity supply of firmsmust cover planned (and in fact here always realized) investments as well asdividend payments not backed up by sales, i.e., the amount p(S� I). Ofcourse, we get additional funds for these investment plans, and thus areduction in the amount of equities needed to finance them, if sales Yexceed demand expectations Y. This procedure guarantees that equitiesdemanded and supplied must be equal to each other if equation (6.74) isagain assumed to hold (see (6.75) for the resulting presentation of thedisequilibrium on the market for goods). Equation (6.76) simply gives thedefinition of aggregate demand — partly based on sales expectations now


(see again (6.53)), and it is easily calculated from it that the differencebetween expected and actual demandmust always be equal to total savingsminus net investment in this model.Equations (6.77) define how firms calculate their demand for new inven-

tories. The first equation states that the desired stock of inventories is apositive fraction of current sales expectations. The second one then addsthat the desired change in inventories I is given by the sum of a trendcomponent (here identified with natural growth for simplicity) and a termwhich says that inventories are further adjusted in order to remove theobserved discrepancy between desired inventories N and actual ones Nwith a time delay of 1/�

�. Note here that there must be inventory invest-

ment in a growing economy even if there is always equilibrium on themarket for goods, here represented through a trend term of the simplestavailable type.In an economy with inventory investment, the level of production is of

course determined by expected sales and this additional inventory invest-ment, i.e., equation (6.78). Equation (6.79) in addition describes how firmsform their demand expectations, here simply in the form of adaptiveexpectations, again augmented by a term which refers to trend growth in astraightforward way. The final new equation of this model is (6.80), whichdescribes how the stock of inventories is actually changed, namely, by thedeviation of actual production from actual aggregate demand (in thisKeynesian setup always equal to planned aggregate demand) which isalways equal in size to the sum of unintended and intended inventorychanges.This concludes our description of the differences of this Keynes—Metzler

model from the previous dynamic multiplier version of it. In the nextchapter we will augment the money-wage Phillips curve by a term repre-senting the rate of capacity utilization of the employed workforce (as ameasure of their over- or undertime work��). Similarly, one can augmentthe price Phillips curve by a term representing the influence of inventorydisequilibrium on price adjustments, like ��

��(N�N)/K . This is a

meaningful extension of the considered price dynamics which, however,will not be investigated within the scope of this book and which generallyhas neither yet been investigated in the applied literature on price Phillipscurves (see Fair 1997a,b in particular).In the general case of an endogenous determination of taxes per unit of

capital and the existence of a feedback mechanism of government debtaccumulation B(t) on the rest of the dynamics, we get from calculationssimilar to those in the case of the basic Keynesian prototype (4.24)—(4.28)

�� Which introduces variable work-time into the Keynes—Metzler model.


the following autonomous, now seven-dimensional, dynamical system inthe variables � �w/p, l�L/K, m�M/(pK), �, b�B/(pK), y�Y/K,and ��N/K:

�� [(1� �)��(V�V� )� (

�� 1)�

�(U�U� )], (6.84)

l1 � n� i( · )�� i�(�� r��)� i

�(U�U� ), (6.85)

m� �� n� [�

�(U�U� )�

��(V�V� )]� l1 , (6.86)

�� [��(U�U� )� ��(V�V� )]��(�

� n��), (6.87)

y� � ny��(y� y)� i( · )y, (6.88)

�� y� y� i( · )�, (6.89)

b� � (��

�)m� (�� n)b� [ (�

�(U�U� )�

��(V�V� ))� l1 ]b. (6.90)

As new relationships we now have for output y�Y/K, in contrast to thedetermination of aggregate demand y�Y/K, the expression

y� (1� n��)y��

�(��y� �), (6.91)

y��y/x� (1� s�)(�� t�)� i

�(�� r��)

� i�(U�U� )� n� �� g, (6.92)

and again the mostly standard abbreviations (r� � r�)

V� l/l,U� y/y�, l�L/K� y/x (y not const.!),�� y� � ��l� y� � ��y/x,r� r

�� (h

�y�m)/(h

�(1� �)),

t�T/K� �(�� rb), t�� t� rb,g� t��

�m,

s( · )� s�(�� t�)� (g� t�)�K1 � I/K� i( · )� n.

6.3.2 The dynamics of the private sector

Let us now assume for the remainder of this section’s treatment of theKeynes—Metzler model type our standard rule tn � t � rb � const. for thecollection of lump-sum taxes. We also assume the simpler rule g� G/K� const. for the determination of the level of government expendituresin the place of g� t��

�m we have used in the general description of the

model. This alternative fiscal policy rule induces only very minor changesin the model’s dynamic behavior. Furthermore, we set U� �V� � 1 fornotational simplicity. This then gives an integrated six-dimensional dy-namical system in �, l, m, �, y and � with an appended b� dynamics, sincethe influence of b on S

�/K (and thus on y) is now suppressed. The system


under discussion may be written

�� [(1� �)��(V� 1)� (

�� 1)�

�(U� 1)], (6.93)

l1 �� i�(�� r� �)� i

�(U� 1), (6.94)

m� �� n� [�

�(U� 1)�

��(V� 1)]� l1 , (6.95)

�� [��(U� 1)� ��(V� 1)]��(�

� n��), (6.69)

y� ��(y� y)� l1 y, (6.97)

�� y� y� (l1 � n)�, (6.98)

where we continue to use the abbreviations

y� (1� n��)y��

�(��y� �),

y��y/x� (1� s�)(�� t�)� i

�(�� r��)� i

�(U� 1)

� n� �� g,

and

V� l/l,U� y/y�, l�L/K� y/x,�� y� � ��y/x, r� r

�� (h

�y�m)/h

�.

The system represented by equations (6.93)—(6.98) is a Keynesian goods-market disequilibrium, money-market equilibrium growth model which isbased on delayed quantity adjustment as well as a sluggish wage- andprice-level adjustment. This model thereby generalizes considerably a var-iety of its limit cases with partially infinite adjustment speeds known fromthe literature. We stress here again that the model’s structural equationshave been chosen as simply (i.e., as linearly) as possible in order to clearlyseparate its basic dynamical structure from additional complexities arisingfrom more refined behavioral relationships (which may be needed subse-quently in order to make the model economically viable). This in particularshows that there are always some ‘‘natural’’ nonlinearities involved in theconstruction of amodel of Keynes—Metzlermonetary growth because of itsgrowth rate formulae and because of some multiplicative (and similar)expressions in the state variables of the model.There is a unique steady-state solution or point of rest of the dynamical

system (6.93)—(6.98) fulfilling ��, l�,m

�� 0 which is given by the following

expressions

y�� y�, l

�� l

�� y

�/x, y

�� y

�� y

�/(1� n�

��), (6.99)

m�� h

�y�, (6.100)

��

�� n, (6.101)


��g� t�� n

s�

� t�, (6.102)

��y��

�l�

, (6.103)

r��

�� n, (6.104)

��

��y�. (6.105)

We assume that the parameters of the model are chosen such that thesteady-state values for �, l, m, �, and r are all positive.

Proposition 6.4: Consider the Jacobian J of the dynamical system(6.93)—(6.98) at the steady state. The determinant of this 6 � 6matrix det J, isalways positive. It follows that the system can only lose or gain asymptoticstability by way of a Hopf bifurcation (if its eigenvalues cross the imaginaryaxis with positive speed).

Proof: Since proportionality factors with positive signs do notchange the sign of det J, we can replace all growth rates by time derivativeson the left hand side of dynamical system (6.93)—(6.98) without modifyingthe sign of det J. Moreover, parts of rows of J which are proportional toother rows of J can be suppressed without any change in det J. As far as thecalculation of the sign of det J is concerned, it suffices therefore to considerthe reduced system

�� const.V� const.U,l�� i( · )� i

�(�� r��)� i

�U,

m� �� const.U� const.V,�� const.U� const.V� const.�,y� � const.(y� y),�� y� y,

where the expressions ‘‘const.’’ always stands for positive magnitudes.Proceeding in this way, one can simplify further�� to obtain

�� const.V� const.y/l,l�� i

�(�� r��),

m� � � const.U�� const.y,�� const.�,y� �� const.�,�� y� y.

�� Since the constants in the third and fourth equation are the same and sincey� y� � t�� s

�(�� t�)� i( · )� g.


In this way, one can finally achieve the following reduced form of thedynamical system (6.93)—(6.98) as far as the qualitative calculation of thesign of det J is concerned (where the constants are of no importance):

�� 1/l, l� ��m,m� �� y�� ,�� , y� � �, �� y.

This last representation of our dynamical system gives rise to the followingreduced sign structure in its Jacobian:

�0 � 0 0 0 0

0 0 � 0 0 0

0 0 0 0 0 �

0 0 0 � 0 0

� 0 0 0 0 0

0 0 0 0 � 0�

It is now a routine exercise to show that the sign of the determinant of thismatrix is positive.�

Let us now investigate some numerical properties of the six-dimensionalgrowth dynamics of Keynes—Metzler type in the neighborhood of theproven Hopf bifurcations. The parameter set for these numerical plots ofthe six-dimensional dynamics is given in table 6.1.Figure 6.1 shows on its left hand side the Hopf bifurcation locus for the

three parameter sets (��,��), (�

�,��), and (��,��), i.e., the locus where a

supercritical, subcritical, or degenerate Hopf bifurcation occurs. A verticalline is used to separate sub- from supercritical Hopf bifurcations. It canalso be seen from figure 6.1 that the parameter set of table 6.1 gives a pointjust above the Hopf loci in (�

�, ��) space as well as (�

�, ��) space and just

below the Hopf curve in (��,��) space (in the middle of the depicted

supercritical domain).Let us consider the (�

�,��) space as an example. For any given �

�,

increasing ��from 0 to 1 means that the system will reach a point where it

loses its stability in a cyclical fashion (at ��). At a supercritical Hopf

bifurcation this will happen via the birth of an attracting limit cycle which‘‘surrounds’’ the now unstable steady state (�

��

�). At a subcritical Hopf

bifurcation an unstable limit cycle (which exists for ��

�,��sufficiently

close to ��) will disappear as �

�approaches ��

�, where the corridor of local

asymptotic stability that existed beforehand has vanished. At a degenerateHopf bifurcation, this same loss of stability need not be accompanied byeither the ‘‘birth’’ of a stable limit cycle (above ��

�) or the ‘‘death’’ of an

unstable limit cycle (below ��), but here purely implosive behavior may


Table 6.1.

s�� 0.8, �� 0.1, y�� 1, x� 2, n� 0.05.

h�� 0.1, h

�� 0.2, i

�� 0.25, i

�� 0.5.

�� 0.21, �

�� 1,

��

�� 0.5.

g� 0.32, r��

�� 0.068625, �� 0.22, �� 0.5.

��

� 0.3, �� 0.2, �

�� 0.75,

�� 0.05, t�� 0.32.

simply change into a purely explosive one. These various types of Hopfbifurcations are treated and depicted in their details in, for example,Wiggins (1990, ch. 3). The (�

�,��) diagram in figure 6.1 thus basically shows

that there is generally — up to very small parameter values of ��— the birth

of a limit cycle as ��crosses the depicted Hopf locus.

The same occurs in the next bifurcation diagram for ��, the adjustmentspeed of inflationary expectations in the place of �

�, the adjustment speed

of wages. This figure in addition shows that a choice of the parameter ��,

the adjustment speed of prices, sufficiently small will make the six-dimen-sional dynamical system locally unstable. The two plots considered suggestthat flexible wages and inflationary expectations and very sluggish priceswork against local asymptotic stability. We will return to this questionwhen the six-dimensional dynamical system is decomposed into threetwo-dimensional dynamical systems in the next subsection.The last Hopf-bifurcation diagram is for the two adjustment speeds of

the Metzlerian inventory mechanism, i.e., ��and �

�, the speed of adjust-

ment of sales expectations and of planned inventory adjustments towardsdesired inventory stocks. It shows that there here exists a band of stablesteady states, limited by a region of unstable steady states for low values of��and �

�as well as for high values of these parameters. Moreover, loss of

stability via increased ��is always ‘‘subcritical,’’ while loss of stability via a

decreased ��may be sub- or supercritical (as shown in the diagram). Again,

sufficiently low or high adjustment speeds here work against local asym-ptotic stability, with respect to both �

��and �

�.

Hopf bifurcations have generally been considered for two-dimensionaland three-dimensional systems in the economics literature. A six-dimen-sional system like the one above is surely much more demanding withrespect to an analysis of the complete set of the Routh—Hurwitz stabilityconditions and at present out of reach for this system. Yet, even fortwo-dimensional systems (and evenmore so for three-dimensional systems)it is generally a horrendous analytical task to investigate whether the Hopfbifurcation is sub- or supercritical, and a proof of this is therefore generallymissing in applications in the economics literature (exceptions are Lux


Figure 6.1 Hopf bifurcation curves, stable limit cycles (projections), or stablecorridors


1993, 1995). Numerical methods therefore have to be used in all such casesin order to decide on the important part of the Hopf theorem, the existenceof either stable limit cycles (describing persistent oscillations) or unstableones (determining stability corridors).On the right hand side of figure 6.1 we show in addition an example of a

stable limit cycle, generated via a supercritical Hopf bifurcation. Note thatthis stable limit cycle is generated solely by the intrinsic nonlinearities ofour Keynes—Metzler monetary growth model, and not by nonlinearityassumptions on the behavioral or technical relationships of this model. Thefigures to the right show how the limit cycle is approached when the steadystate of the model is disturbed via a small l shock. We here show the (�, l),(m,�), and (y, �) projections of this limit cycle. These projections will becompared with the corresponding two-dimensional decompositions of thesix-dimensional dynamics in the next section.

6.3.3 The real, the monetary, and the inventory subdynamics

(a) The real wage and accumulation dynamicsIn order to isolate the real dynamics (�, l) from the rest of the system, wemake the following set of assumptions:

(1) ��

��,�

�� 0: y� y� y, �� 0, i.e., goods market equi-

librium with no inventories,(2) h

�� : r� r

�, ��

�� n, i.e., the liquidity trap at the

steady state and long-run steady-state inflationary expectations.

In particular, the second set of assumptions is of course only justified froma mathematical and methodological point of view. As before, assumingt�� const., allows us to ignore the dynamics of theGBRwhen studying the�, l subsystems (the same holds for the other subsystems to be consideredbelow). The resulting two-dimensional dynamical system turns out to be

�� [(1� �)��(V� 1)� (

�� 1)�

�(U� 1)], (6.106)

l1 �� i�(�� r

��

�)� i

�(U� 1)� n� g� t�� s

�(� � t�),

(6.107)

with U� y/y�,V� (y/x)/l,�� y� �� (y/x)�.The value of y�Y/K has now to be calculated from the following goods

market equilibrium condition (see (6.92)):

y��y/x� (1� t�)� i�(�� r

��

�)� i

�(U� 1)� n� � � g.

(6.108)

At the steady state of the dynamical system (6.106)—(6.107), we have


�� t�� (n� g� t�)/s

�(via l1 � 0) and i

�( · )� 0 via r

��

��

�. Em-

ploying again l1 � 0 then gives y�� y� and thus l

�� y

�/x due to �� 0. The

steady-state value of � finally is given by definition of � as ��

(y��

�)/l

�, as in the six-dimensional case.

The goods-market equilibrium condition (6.108) thus reads in the steadystate

y��y�/x� (1� s

�)(�

�� t�)� n� �� g.

Both equilibrium conditions can be brought to the form

[(i�� s

�)(1��/x)� i

�/y�]y� t�� s

�t�� i

�(r��

�)

�i�� (i

�� s

�)�� g� n,

which gives

y�[(i

�� s

�)(1��

�/x)� i

�/y�]y�

(i�� s

�)(1��/x)� i

�/y�

�(i�� s

�)(1��

�/x)y�� i

�(i�� s

�)(1��/x)y�� i

�

y�. (6.109)

The properties of the function y(�) have already been discussed in chapter4, and gave rise there to three different situations for the above dynamics.One of these cases will be excluded here from consideration by way of theassumption

Z� (i�� s

�)(1��

�/x)y�� i

�� 0.

This restricts the set of admissible parameters i�, i�, s�� such that �� (�)� 0

holds true, whenever the function

�(�)�Zy�

(i�� s

�)y�� i

�/(1� �/x)

� �,

is well defined. The dependence of the rate of profit � on the real wage rate� is therefore the conventional one in our remaining cases. Nevertheless,the sign of y�(�) will be ambiguous at and around the steady state, sincethere follows in this case:

sign y�(�)� sign (i�� s

�)

The real dynamics (6.106)—(6.107) therefore allows, even under the above

� (1��/x)y��

n� g� (1� s�)t�

s�

.


assumptionZ� 0, for two very different situations of the dependence of y,U, and V on the real wage �.

Proposition 6.5: The steady state of the dynamical system (6.106)—(6.107) is locally asymptotically stable (unstable) in case 1: y�(�

�)� 0 if �

�is

chosen sufficiently high and ��sufficiently low (�

�sufficiently low and �

�sufficiently high). The opposite statements apply to case 2: y�(�

�)� 0.

Proof:The Jacobian J of (6.106)—(6.107) at the steady state is givenby

J�� [(1�

�)��V�� (

�� 1)�

�U�]� (1�

�)V��

�s�� 0 � ,

with �� 0 and V�� 0, i.e. det J� 0 always.� The assertions then im-

mediately follow from V�,U� � 0 (case 1) and V�,U� � 0 (case 2) via thetrace of the matrix J.�

Proposition 6.6: The Hopf bifurcation locus in (��, ��) space is

given by the straight line:

��

�1�

�1�

�

��.

Proof: This locus is defined by trace J� 0 (det J� 0!), whichyields the assertion, sinceU�� y�(�)/y�,V�� (y�(�)/x)/l

�and l

�� y

�/x.�

For y�(�)� 0 the system is locally stable above the line in proposition 6.6and unstable below it. The opposite is true for case 2: y�(�)� 0.As already stated, there is no easy analytical procedure by which we can

determine the character of the Hopf bifurcation that is taking place along��(��) (see Wiggins 1990, p.277, for details). Numerical investigations as in

the preceding section, however, show a potential for supercritical Hopfbifurcations (as in the six-dimensional case). With respect to the basicparameter set of table 6.1 of the preceding section, we get for the two-dimensional dynamics ��

��

�and y�(�)� 0 (i

�� 0.25� s

�� 0.8).

We thus get the ‘‘surprising’’ result that wage flexibility destabilizes thesix-dimensional system (the opposite holds for price flexibility), while theopposite is true for the two-dimensional limit case. In this respect, thetwo-dimensional case gives completely misleading information on thestability properties of the six-dimensional case.

� In the excluded case we have det J� 0 and thus a saddlepoint.


If unstable, the two-dimensional system can be made globally stable in atailored domain by assuming a nonlinearity of tanh type in the investmentfunction (as in Kaldor’s 1940 trade cycle model), and a nonlinearity of tanhtype in the �

�(V� 1) Phillips curve of the model if y�(�)� 0 holds (as in

Rose’s 1967 employment cycle model). In this case, the model generates anemployment growth cycle as in Rose (1967) via an application of thePoincare—Bendixson theorem (see chapter 4 for details and also for theconsideration of further subcases, for example, of the Goodwin 1967growth cycle type). The real sector of our six-dimensional economy there-fore exhibits features of a ‘‘Classical growth cycle’’ (��(�)� 0,K1 (�)� 0) incombination with Keynesian aggregate demand problems (y�(�)� 0).In sum, we may here in particular state that increased wage flexibility

will be destabilizing in the case where goods-market equilibrium y re-sponds positively to changes in income distribution (changes in the realwage), since real wage increases then increase employment, and thus theupward pressure on nominal wages (and also on real wages, for a givendegree of price-flexibility). By contrast, increased price flexibility will bestabilizing in this case. If unstable, the dynamics can be made economicallyviable by introducing certain nonlinearities that are well known from theliterature on trade and growth cycles.

(b) The monetary dynamicsTo isolate the monetary dynamics from the rest of the system we make thefollowing set of assumptions:

(1) ��

�� ,�

�� 0: y� y� y, �� 0, i.e., goods market equi-

librium with no inventories,(2) �

�� 0,

�� 1:��

�(�� 0), i.e., no real wage dynamics,

(3) l1 � 0: l� l�, i.e., no variation in relative factor proportions (medium-

run analysis).

We thus in particular exclude the Goodwin—Rose growth cycle fromconsideration. Note, here, that investment is allowed to vary in the presentcase without having any capacity effects different from the growth rate n ofthe labor force.The above assumptions lead to the monetary subdynamics

m� �� n��

�(U� 1), (6.110)

�� (U� 1)� ��(�

� n� �), (6.111)

where U� y/y�� 1 and where equilibrium output y is now given by

y��y/x� (1� s

�)(�� t�)� i

�(�� r��)� i

�(y/y�� 1)

� n� � � g, (6.112)


with � � y(1��/x)� �, r� r

�� (h

�y�m)/h

�.

Making use of the steady-state values of m and �, equation (6.112) can betransformed to

y� y��(i�/h

�)(m�m

�)� i

�(��

�)

h�i�/h

�� (s

�� i

�)(1��

�/x)� i

�/y�

� y��1�(i�/h

�)(m�m

�)� i

�(��

�)

h�i�y�/h

�� (s

�� i

�)(1��

�/x)y�� i

��

� y��1�(i�/h

�)(m�m

�)� i

�(��

�)

h�i�y�/h

��Z � , (6.113)

and we recall that Z has been defined below equation (6.109). Viewed fromthe perspective of the preceding subsection on the real subdynamics(h

��), the related case h

�� (but large) gives a negative denominator

in (6.113) under the assumption we have made there, and therefore givesrise to a function y(m,�) with y

�� 0, y�� 0. With respect to conventional

macrostatics, these two partial derivatives represent an abnormal Keynesand a negative Mundell effect on effective demand. This is so since anincrease in real balances (via a decrease in the price level p) is thencontractionary and an increase in � does not stimulate investment andeffective demand, but rather will reduce the latter. Both effects will becomepositive (y

�,y� � 0), and are regarded as normal if h

�is decreased, s

�increased (relative to i

�) and i

�decreased to a sufficient extent.

Solving m� 0, �� 0 for the unknown steady-state values �� n�

��,U

�� 1 gives �

��

�� n,U

�� 1, i.e., y

�� y�. Equation (6.113) then

implies r��

��

�� y�(1��

�/x)� �� n and thus m

�� h

�y� for

our second dynamic variable in the present situation. The steady state ofthe dynamical system (6.110)—(6.111) is therefore once again the one that isgiven by the six-dimensional dynamical system (6.93)—(6.98). With respectto this steady state we can now state the following.

Proposition 6.7: (1) The dynamical system (6.110)—(6.111) exhibitssaddlepoint behavior around the steady state if Q� h

�i�y�/h

��

(s�� i

�)(1��

�/x)y�� i

�� 0 holds and exhibits a positive determinant of

its Jacobian if Q� 0; (2) the Hopf locus in (��,��) space of the latter case is

given by �� y�/�� y��m�y�/y� , with y�� (i

�y�/h

�)/Q, y�� i

�y�/Q,

i.e., m�y�/y�� h

�y�/h

�. This locus is therefore a decreasing function of the

parameter ��; (3) the dynamical system (6.110)—(6.111) is locally asymptoti-

cally stable below this locus and unstable above it.

Proof: (1) The Jacobian of (6.110), (6.111) at the steady state reads


J��

�U�m �( �

�U� � 1)m

�� U�

�� U�� .

In the case Q� 0 we have U�,U�� 0 and the opposite signs hold for

Q� 0. For det J we have in general

det J� ��

�U�m �m

�� U�

�� U�m(�� ).

Hence det J is negative in the first case and positive in the second.(2) In the second case, the Hopf locus is characterized by trace J� 0 (det

J� 0!), which gives �� U��

�U�m, from which the result in

part (2) follows.��(3) This result follows from the observation that �� implies trace

J� 0 (and trace J� 0 in the case �� ).�

For the basic parameter set (table 6.1) of the six-dimensional model of thepreceding subsection we have

Q� 0.1 · 0.25 · 1/0.2� 0.55 · 0.5� 0.5�� 0.15� 0,

and thus obtain a saddlepath dynamics for the two-dimensional case. Yet,det J� 0 in the six-dimensional case, i.e., the saddlepath behavior does notshow up in the general model in an obvious way. We have exemplified inthe preceding section the Hopf locus of the (�

�,��) space (see figure 6.1).

There is no equivalent to this Hopf locus in the two-dimensional case. Wehave seen above that the case Q� 0 gives rise to a Hopf locus in (�

�, ��)

space, but now with negative slope in the place of the positive slopeobserved in the six-dimensional case. Fast adjustment of inflationary ex-pectations via ��, i.e., giving a high weight to past inflation in comparisonto future steady-state inflation, leads to local instability (and in fact to thebirth of stable limit cycles via supercritical Hopf bifurcations, as one canshow numerically). Appropriate nonlinearities in the investment functioni( · ) should again allow for global limit cycle results via the Poincare—Bendixson theorem.

(c) The inventory dynamicsIn this case, we combine the assumptions h

��

(r� r�,� ��

��

�� n) with the assumptions �

�� 0,

�� 1(��

�,

�� 0) and K1 �L1 � n(l� l�, l1 � 0). This gives rise to the isolated sales

expectations and inventory dynamics

y� ��(y� y), (6.114)

�� Note that ��is constant in the case �� 0.


Figure 6.2 Hopf bifurcation loci of the inventory cycle for Z� 0

�� y� y� n�, (6.115)

with

y� y� n��y��

�(��y� �),�� y� ��y/x

and

y� y� (i�� s

�)�� i

�(y/y�� 1)� t�(1� s

�)� i

�(r��

�)

� n� g� [(i

�� s

�)(1��

�/x)� i

�/y�]y� [i

�/y�

� (i�� s

�)�

�/x](y� y)� const.

�Zy�Z�(y� y)� const.,

where we set Z�� i

�/y�� (i

�� s

�)�

�/x.

At the steady state of this dynamical system we have y�� y

�and

y�� y

�� n�

�. Moreover, � ��

�, r� r

�, and ��

�imply y

�� y� and

y�� y

�� y

�/(1� n�

��), i.e., the steady state of (6.114)—(6.115) is again the

one implied by the steady-state solution for the six-dimensional dynamics.


In order to calculate the sign of the determinant of the Jacobian of(6.114)—(6.115) at the steady state, it suffices to consider the system

�y� y

y� y� or �Zy

y� y� ,if the natural rate of growth n is chosen sufficiently close to zero. This righthand reduction of the original dynamics now immediately shows that det Jis of the same sign as�Z. Under the assumptions of the subsection on thereal dynamics, we therefore get that the inventory dynamics (we havesuppressed in that subsection) is unstable and of saddlepoint type.In the opposite case of a negative value ofZ (and thus det J� 0), we have

to investigate the trace of J in addition, i.e., the expression (for n sufficientlyclose to zero) trace J� �

��(Z�Z

��)��

�(Z

�� 1).

This expression gives rise to the following proposition.

Proposition 6.8:AssumeZ� 0 and the rate of growth n sufficientlysmall. (1) The steady state of (6.114)—(6.115) is locally asymptotically stablefor all �

�sufficiently small; (2) the steady state of (6.114)—(6.115) is unstable

if ��

� (1�Z�)/(Z

��) and �

�sufficiently large; (3) the Hopf bifurcation

locus is given by ��

� �Z/(Z��

� (Z�� 1)/�

��)� 0 if �

��

� 1� 1/Z

�[Z

�� 0!].

Proof: (1), (2) Obvious from the expression trace J�

��Z��

�[��Z

��

� (Z�� 1)].

(3) The result follows since Z� 0 i�� s

�(since 1��

�/x�

0)Z�� 0(�Z� (s

�� i

�)), i.e., ��

�� 0 if �

��is chosen sufficiently

large.�

Franke and Lux (1993) introduce a flexible stock adjustment principlewhich allows them to show global stability for a Metzlerian model ofsimilar type by means of the Poincare—Bendixson theorem.Figure 6.2 shows the Hopf locus for the three cases Z

�� 1, Z

�� 1, and

Z�� 1 to which this situation can give rise.��

6.3.4 Conclusions

In the preceding subsectionwe have considered three isolated subdynamicsof the general six-dimensional dynamics by means of appropriate limitcases of the six-dimensional case. With respect to the parameter set of table6.1 one can easily check that the real dynamics is then characterized by

�� A��Z/y�/(Z��),B� (1�Z

�)/(Z

��).


Figure 6.3 Hopf bifurcation curves, stable limit cycles, and stability corridors forZ� 0


Z� 0.225� 0(��(�)� 0, y�(�)� 0), i.e., it falls into the range of cases con-sidered in the subsection on the real dynamics. In particular, wage flexibil-ity is stabilizing and price flexibility is destabilizing in this two-dimensionalcase. The opposite is, however, true in the general six-dimensional casewith respect to the role the parameters �

�and �

�play. Furthermore, the

monetary dynamics is of saddlepoint type for this parameter set, sinceh�i�y�/h

��Z� �0.1� 0 holds true (y

�, y� � 0!). Yet, in the correspond-

ing six-dimensional case we have found by numerical investigations that aHopf bifurcation locus is then present and that an increasing parameter �

�will stabilize the dynamics. Finally, the two-dimensional Metzler systemalso exhibits a saddlepoint with respect to this basic parameter set, whereasa band of local asymptotic stability existed in the six-dimensional case,surrounded by super- or subcritical Hopf bifurcation loci. We concludethat the two-dimensional limit cases (which represent the well-establishedpartial dynamical models of Goodwin—Rose, Tobin, and Metzler type inthe literature) are not very revealing with respect to the properties of thegeneral six-dimensional dynamics.Hence, further possibilities for analyzing such integrated high-dimen-

sional dynamical systems have to be found in order to gain more insightinto the interaction of dynamics of the Goodwin—Rose, Tobin, andMetzlertype. It is possible to decompose the six-dimensional case in still differentways, for example, into an�,m, � subsystem and an l, y, � subsystemwhichcan still be treated analytically via the Routh—Hurwitz and Hopf theorems(see chapter 4 for an example). In special cases, five-dimensional systemsare also within reach, as we have seen in the case of the KT model. Yet, ingeneral, we have to rely on numerical methods in order to get a properfeeling for the dynamic implications allowed by the general six-dimen-sional system of Keynes—Metzler type.The Hopf cycle of the real dynamics can be combined with a Hopf

situation for the monetary one if h�is decreased sufficiently (whereby

interest-rate flexibility is increased). Reducing h�from 0.2 to 0.1 implies

h�i�y�/h

��Z� 0.025 and det J� 0(y

�, y�� 0) for the monetary two-

dimensional dynamics whereby a positive Hopf locus can now be found inits (�

�,��) parameter space. Yet, due to Z� 0, the Metzlerian two-dimen-

sional dynamics stays of saddlepoint type.In the caseZ� 0, finally, we get Hopf loci for the latter two-dimensional

case (and the monetary dynamics), but saddles for the real dynamics, i.e.,there is no situation where all two-dimensional dynamics allow for Hopfloci simultaneously. This is in striking contrast to the six-dimensionalsituation, whose typical Hopf loci are exemplified in figure 6.2. This figureshows that there can be considerable switches in stability regimes on thetwo-dimensional level. Yet, as the figure 6.2 demonstrates in comparison to


figure 6.1, no such significant change is visible on the six-dimensional level.It seems that the six-dimensional dynamics is much more robust than itsthree isolated two-dimensional subcases.

6.4 A (5� 1)-D modification of the six-dimensional Keynes–Metzlermodel

In this section we continue the analysis of the preceding section andinvestigate further the dynamic properties of a general Keynesian monet-ary growth model, with a conventional IS—LM block, based onMetzleriangoods-market disequilibrium adjustment processes in the place of theconventional static or dynamic multiplier approach.Quantities on the goods market adjust through an inventorymechanism

based on sales expectations and planned vs. actual inventory changes.Corresponding to this sluggish adjustment of quantities there are alsosluggish price and wage adjustments, in a basically conventional way ofexpectations augmented wage or price Phillips curves with demand press-ure and cost-push components. These real and nominal adjustment pro-cesses are supplemented by a money-market equilibrium equation as the-ory of the nominal rate of interest.These are the essential building blocks of the model which is again made

complete by the specification of the budget equations of households, firms,and the government, together with some additional details. The structuralequations of the model introduced below will differ in some minor respectsfrom those used in the preceding section, making the model from a math-ematical point of view less intertwined and downplaying the role of theMetzlerian inventory process to some extent. Such a simplification pro-vides an intermediate step between the Kaldorian and the Metzlerianmodels considered in the two preceding sections. By introducing as usualappropriate state variables in intensive form, the model can be reduced to anonlinear autonomous differential equation system of dimension six with,however, only five state variables that are really interdependent.Through its construction, the model again integrates three important

partial (two-dimensional) views on the working of the macroeconomywhich we have investigated in detail in the subsections of the precedingsection. These investigations will lead to slightly different results for thepresent model, due to the simplifications proposed below for the Keynes—Metzler model, but will not be repeated here, since these differences canbasically be neglected in what follows.We have seen that the results obtained from these prototypic sub-

dynamics will generally not be characteristic for the integrated system,which severely limits the value of these partial analyses. Flexibilities that


work against economic stability on the two-dimensional level can work infavor of it on the six-dimensional level and vice versa, as we shall see againbelow. Finally, as we shall show, complex behavior can occur that is notpossible on the two-dimensional level. We conclude that the use of partialmodels that separate growth from inflation and from inventory adjustmentmay be very misleading with respect to the implications they have forstability, types of fluctuations, and economic policy when compared withthe results that their interaction generates.When we come to consider high speeds of adjustment for prices or

quantities in this model type, we will find again that generally the viabilityof such naturally nonlinear models is destroyed. It then becomes obviousthat important nonlinearities due to changing economic behavior far offthe steady state of the model are still lacking. After providing a list of themost basic quantity or value constraints that may come into play insituations of larger business fluctuations, we subsequently choose one (andonly one) particular type of behavioral nonlinearity in order to attempt torestrict the explosive nature of the dynamics for higher adjustment speeds.This nonlinearity concerns a basic fact of the postwar period, namely, thatthere has been no deflation in the general level of wages even in periods ofhigh unemployment. The wage inflation Phillips curve of the model, whichgenerally operates in an inflationary environment, is thus modified suchthat no decrease in the nominal wage level can become possible. Thissimple change in the model’s dynamics, the exclusion of nominal wagedeflation, has dramatic consequences for its viability (and also its complex-ity). Thus, Keynes’ (1936) judgment onworkers’ behavior and its stabilizingconsequences finds a striking (numerical) illustration in this section in thecontext of a general Keynesian model of monetary growth dynamics.

6.4.1 A simplified Keynes—Metzler model of monetary growth

The following model structure represents a somewhat simplified version ofthe model type considered in the preceding section, more stress is nowbeing laid on mathematical simplification in the place of full economicinteraction.� In contrast to the six interdependent state variables of thepreceding section, the last state variable of the present model will not feedback into the first five laws of motion of the model. Nevertheless, withrespect to economic content the model is still very close to that of thepreceding section. Its economic motivation will therefore be discussed hereonly briefly.The first change concerns the household sector, and in particular its

� See also Chiarella and Flaschel (1996b), Chiarella et al. (1999) for related presentations ofthis model type and its theoretical and numerical properties.


money demand function

M� h�pY� h

�pK(r

�� r). (6.116)

Money demandM is now specified as a function of the nominal value ofexpected sales pY (as a proxy for expected transactions) and of interest rater in the usual way. The form of this function has been chosen such that itallows for a simple linear formula for the rate of interest in terms of the statevariables of the model.The second and final change concerns the description of the behavior of

firms, which is now based on expected sales throughout. Thus


L�Y/x,x� const.,V�L/L�Y/(xL). (6.118)

Firms expect to sell commodities in amount Y and produce them in thetechnologically simplest way possible, namely, the fixed proportions tech-nology characterized by the normal output capital ratio y��Y�/K and afixed ratio x between expected salesY and laborL needed to produce thisoutput. This simple concept of technology allows for a straightforwarddefinition of the rates of utilization U and V of capital and labor.Note here, however, that firms may produce more or less than expected

sales, depending on their inventory policy. In order to suppress somesecondary economic feedback effects we have nevertheless assumed thatthe economic actions of firms are based on a measure of capacity utiliz-ationU as defined above, and that they pay their workforce on the basis ofthe employment generated by expected sales, while planned changes ininventories are accompanied by over- or undertime work of the employed(that does not show up in the wage bill).There are thus only three changes made in the Keynes—Metzler model

we have considered previously. The present model again integrates theinteraction between real wages and capital accumulation, between infla-tion and the expected rate of inflation, and between expected sales andactual inventory levels, the latter, however, in a less complete way than inthe preceding section.The above general model of Keynesian monetary growth dynamics can

be reduced to the autonomous six-dimensional dynamical system in thevariables �, l, m, �, y, and �,

�� [(1� �)��(V� 1)� (

�� 1)�

�(U� 1)], (6.119)

l1 ��i�(�� r��)� i

�(U� 1), (6.120)

m� �� n� [�

�(U� 1)�

��(V� 1)]� l1 , (6.121)

�� [��(U� 1)� ��(V� 1)]��(�

� n��), (6.122)


y� ��(y� y)� l1 y, (6.123)

�� y� y� (l1 � n)�. (6.124)

For output per unit of capital y and aggregate demand per unit of capital ywe have the expressions

y� (1� n��)y��

�(��y� �), (6.125)

y��y/x� (1� s�)(�� t�)� i

�(�� r� �)

� i�(U� 1)� n� � � g

� y� (i�� s

�)�� i

�(r� �)� i

�(U� 1)� const. (6.126)

Furthermore, we make use of the abbreviations

V� l/l� y/(lx),U� y/y�, (6.127)

�� y(1��/x)� �, (6.128)

r� r�� (h

�y�m)/h

�. (6.129)

This reformulation of the model shows that the variable � does not appearon the right hand side of the first five laws of motion. It is thus of secondaryimportance in the following. Note that the variable g is here again assumedas being given exogenously.There is a unique steady-state solution or point of rest of the dynamical

system (6.119)—(6.124) fulfilling ��, l�,m� 0 which is given by

y�� y

�� y�, l

�� y

�/x, y

�� (1� n�

��)y

�, (6.130)

m�� h

�y�, �

��

�� n, (6.131)

�� t��

g� t�� n

s�

, r��

��

�� n, (6.132)

��y��

�l�

, ��

��y�. (6.133)

6.4.2 Dynamical properties of the integrated dynamics

Let us now immediately turn to the investigation of the full (5� 1)-Dsystem and reduce the comparison of its features with the three two-dimensional subsystems considered in the preceding section to some nu-merical observations in this section.

Proposition 6.9: Consider the Jacobian of the dynamical system(6.119)—(6.124) at the steady state. The determinant of this 6 � 6-matrix, detJ, is always positive. It follows that the system can only lose or gain


asymptotic stability by way of a Hopf bifurcation (if its eigenvalues crossthe imaginary axis with positive speed).

Proof: Similar to that of proposition 6.3.�

Proposition 6.10: For the entries in the trace of J there holds:

• J��

� 0, i.e., the Rose effect does not show up in the trace of J;• J

�� 0 as in the corresponding two-dimensional case of real growth;

• J

� 0, due to the Keynes-effect r(p), r�(p)� 0 in the l1 term of the thirddynamical law;��

• J

� 0, due to the forward-looking component in the fourth dynamicallaw;��

• J��

��(�Q/y�)� y

�(Q/y�� s

�(1��

�/x)), where Q has been defined

in the preceding section;��• J

�� y � 0 and J

#�� 0 for i� 1, . . ., 5. It follows from proposition 6.4

that the determinant of the Jacobian of the (independent) five-dimen-sional subdynamical system (6.119)—(6.123) is negative at the steadystate.

Proof: Follows from straightforward calculations of the indicatedelements of J.�

We thus have that the destabilizing (or stabilizing) role of the parameters��, �

�, and �� cannot be obtained by just considering the trace of the

matrix J as is true for the related two-dimensional cases. The determinantbeing positive and the trace of J being basically negative (if �

��is chosen

appropriately), it therefore depends according to the Routh—Hurwitz con-ditions on the other principal minors (of dimension two to four) whetherthe steady state of the considered dynamics is locally asymptotically stableor not.There are, for example, fifteen principal minors of J of dimension two,

three of which are given by the three determinants considered in thetwo-dimensional cases in the preceding section. The calculation of the

�� Note here that the state variable y prevents an immediate impact of theKeynes effect (andits consequences for aggregate demand) on factor utilization ratesU andV, and thus on therate of inflation and the corresponding state variable m.

�� The above remark on the Keynes effect here applies to theMundell effectY�� 0, i.e., there

is no longer a destabilizing influence of the parameter �� present in the trace of theJacobian (as in the two-dimensional case).�� It follows that the system must be locally unstable for values of �

��sufficiently large if

Q� 0, see proposition 6.7, since this adjustment parameter is (besides the alwaysstabilizing parameter ��) the only one among the adjustment speed parameters that showsup in the trace of J.


corresponding Routh—Hurwitz conditions for local asymptotic stability isthus a formidable task (and even more so for the other conditions). It isnevertheless tempting to conjecture that these Routh—Hurwitz stabilityconditions might be fulfilled for either generally sluggish or generally fastadjustment speeds. The following numerical investigations of the model,however, show that nothing of this sort will hold true in general.In the presentation of the general Keynes—Metzler monetary growth

model we have made use of linear relationships as much as possible.Technology, behavioral relationships and adjustment equations were allchosen in a linear fashion. Though nonlinear in extensive form, moneydemandwas chosen such that it gave rise to a linear equation for the rate ofinterest when transformed to intensive form. Yet, certain relationships suchas the wage dynamics must refer to rates of growth in order to make senseeconomically. Furthermore, and quite naturally, there are certain productsof variables involved, such as total wages �L or the rate of employmentL/L. Such occurrences make the model a nonlinear one in a natural orunavoidable way. It is one of our aims in the present section to investigatethemodel’s dynamic properties in this naturally nonlinear form in order tosee to what extent the dynamical behavior so generated represents aneconomically, or at least mathematically, viable one despite the negativefindings obtained in the preceding section from its three prototype subsys-tems. Of course, it is not to be expected that the dynamical behavior isviable for all meaningful parameter constellations. Further nonlinearities,in particular from the supply side, will become operative in a variety ofsituations. Nevertheless it is often not necessary to use nonlinearities inwage adjustment, in technology, in investment, etc., in a first step in orderto get an economically bounded behavior. Where the two-dimensionalcases suggest the use of such additional nonlinearities, the correspondingsix-dimensional situation may nevertheless be asymptotically stable or, ifnot, give rise to limit-cycle behavior over certain ranges of the parametersdue to the presence of the natural nonlinearities just discussed.In the intensive form and with respect to the state variables used in

equations (6.119)—(6.124), there are three types of nonlinearities induced bythe structural form of the model. The first type arises because three of thestate variables give rise to a growth rate law of motion (�, l,m). The secondtype is due to the formulation, in per-capital terms, of two of the statevariables (y, �) giving rise to products of the form l1 y, l1 �. The third typeoccurs because there are natural products or quotients of some of the statevariables in the form V� y/l for the rate of employment V and�� y� � ��y/x for the rate of profit �.Note here that the replacement of the state variable l by the state

variable k� 1/l transforms all nonlinearities into product form. Note


Table 6.2.

s�� 0.8, �� 0.1, y�� 1, x� 2, n� 0.05.

h�� 0.1, h

�� 0.2, i

�� 0.5, i

�� 0.5.

�� 0.16, �

�� 1,

��

�� 0.5.

g� 0.32, r��

�� 0.3875, �� 0.1, �� 0.75.

��

� 0.2, �� 0.75, �

�� 1,

�� 0.05, t�� 0.3.

furthermore that the terms l1 z�� k1 z with z�m, y, � lead to trilinearexpressions in their respective laws of motion.In this representation of the dynamics we have, besides growth rates and

the products just mentioned, nonlinearities present only in the ��( · ) term,

in �, and consequently also in the aggregate demand term y. Up togrowth rate formulations we have thus basically only two types of non-linearities involved in the laws of motion of the system, and they both relateto the Rose subdynamical system of the model. Though these terms re-appear in various places, it may therefore be stated that the presentdynamics is in a weak sense comparable to the Rossler system (one bilinearterm) and the Lorenz system (two bilinear terms), but in a less strict way,due to its higher dimension.Let us now turn to a numerical investigation of the (5� 1)-D dynamics.

We shall employ the basic parameter set displayed in table 6.2 in thenumerical illustrations given below (and we shall subsequently only statethe changes taking place with respect to it at various places).Corresponding to the three subdynamical systems considered in the

preceding section, we here look at the stabilizing or destabilizing role of thepairs of adjustment speeds �

�,��, ��, ��, and �

��,��. The shaded areas in

figure 6.4 show the parameter domains where the six-dimensional dynami-cs is locally asymptotically stable. The boundary of these domains is theHopf-bifurcation locus where the system loses its local asymptotic stability,either by way of a supercritical Hopf-bifurcation (where a stable limit cycleis born after the boundary has been crossed) or byway of a subcriticalHopfbifurcation (where an unstable limit cycle is shrinking to ‘‘zero’’ when theboundary is approached). Along the bifurcation line there also exist degen-erateHopf bifurcations separating super- from subcritical bifurcations.Wehave found in many numerical investigations of the model that the bifurca-tions in the following diagrams are generally of a supercritical nature. Theonly important exception is the bifurcation line on the right hand side ofthe �

��, ��parameter space, where the system again loses stability (at

��

� 4.82) for high adjustment speeds of the parameter ��. Note here that

figure 6.4 also shows the independence of the domain of stability from thestate variable � and the parameter �

�.


Figure 6.4 Six-dimensional bifurcation loci and a limit cycle for h�� 0.2 (Q� 0)

Proposition 6.5:With respect to the choice of parameter values intable 6.2, the steady state of the dynamical system (6.119)—(6.124) is locallyasymptotically stable for a high adjustment speed of prices, a low adjust-ment speed of wages, a low adjustment speed of inflationary expectations,and all inventory adjustment speeds with respect to the given parameterset. Finally, the adjustment speed of sales expectations must be in theinterval (0.98, 4.82), i.e., it should be neither too high nor too low.

The corresponding situation of the three two-dimensional subdynamicalsystems is shown in the small squares in the figure 6.4. We can see that thecombination of an explosive real cycle with (unstable) saddlepoint situ-ations in themonetary and the inventory subsystem gives rise to stability inthe integrated six-dimensional system. Note here that there are no perverseKeynes effects Y

�� 0 or Mundell effects Y�� 0 with respect to the aggre-


gate demand function of the six-dimensional system, in contrast to thecorresponding two-dimensional situation. Furthermore we can state:

Proposition 6.5 (continued): With respect to the choice of par-ameter values in table 6.2, the stabilizing properties of price and wageadjustment in the six-dimensional dynamical system are just the oppositeof those suggested by the two-dimensional dynamical subsystem of the realcycle model.

The partial model thus provides the wrong information concerning animportant policy issue, namely, that of the adequate degree of wage flexibil-ity for economic stability. Sluggish wages work in favor of economicstability, while flexible wages do not.With respect to the parameter �

�, the bifurcation point where local

stability becomes lost is approximately given by ��

� 0.16. The finalpicture in figure 6.4 shows the projection in the � � l plane of the stablelimit cycle that is generated beyond this point at �

�� 0.2. This limit cycle

increases considerably in amplitude when this parameter is increasedtowards �

�� 0.3. Thereafter, the dynamics becomes purely explosive.

In figure 6.4 we consider an example of the situationwhere the monetarytwo-dimensional dynamics (see the �

�vs. �� plot) is of saddlepoint type

(Q� 0, see the subsection on the two-dimensional dynamics in the preced-ing section). In the opposite case, Q� 0, the two-dimensional situationinstead exhibits a Hopf bifurcation line (which is shown in the small squarein figure 6.5; the situation for the other two-dimensional dynamics remainsunchanged). Ignoring very small adjustment speeds of the price level thesix-dimensional dynamics has not changed very much qualitatively by theassumption of a parameter value for h

�that gives rise to Q� 0. Yet, the

domain of stability is quantitatively seen to be significantly increased withrespect to �

�, �� by the possibility of stability for the monetary sub-

dynamics. Note again that price flexibility (starting from an unstablesteady state) can restore stability to the six-dimensional dynamics, but notto the two-dimensional dynamics of the monetary subsystem.In figure 6.5 we also show some effects of parameter changes on the

position of the Hopf bifurcation line. In figure 6.5(a) we can see that anincrease of the parameter �� from 0.1 to 0.4 may increase the stabledomain for wage flexibility. Similarlywe consider in figures 6.5(b) and 6.5(c)a decrease of �

�from 0.16 to 0.1 and a decrease of �� from 0.4 to 0.1,

respectively. The main point shown by these diagrams is, however, thattwo-dimensional explosive situations are combined by the integrated dy-namics in such a way that local asymptotic stability can be obtained.We have pointed above to the ‘‘naturally’’ nonlinear structure of our


(a) (b)

(c) (d)

Figure 6.5 Six-dimensional bifurcation-loci and a limit cycle for h�� 0.8 (Q� 0)

dynamical system. The question arises whether this basically ‘‘bilinear’’system allows for a period-doubling sequence towards complex dynamicsas, for example, the Rossler system with its single bilinear term (see forexample Strogatz 1994, p.377, for a graphical presentation in the case ofthis system). Figure 6.6 provides such an example for the dynamical systemof this section and the basic parameter set given above (but withh�� 0.08, �� 0.4).Figure 6.7 shows the kind of attractor that may be generated from such a

sequence of period-doubling bifurcations of the limit cycle that is generatedby the Hopf-bifurcation parameter �

�. Note that all these figures represent


Figure 6.6 A period-doubling route to complex dynamics (h�� 0.08, �� 0.4)

projections of the dynamics that is taking place in six-dimensional phasespace onto the �� l plane.These numerical simulations also show that the cycle generated in this

way becomes larger and larger. In particular, it by no means stays in aneconomically meaningful subset of the phase space. Increasing the par-ameter �

�further than shown above will eventually also destroy math-

ematical boundedness. From an economic point of view, it is thus clear thatadditional forces must come into being when certain ceilings or floors areapproached with respect to quantity or value magnitudes. This is the topicof the following subsection.


Figure 6.7 At the edge of mathematical boundedness (h�� 0.08, �� 0.4)

6.4.3 The case of no nominal wage deflation

When the fluctuations generated by the naturally nonlinear model of thissection become very large, as in the situation shown in the figure 6.7, oreven unbounded, they may or will leave the domain of economicallyadmissible values. Then, or even long before such a point is reached, othereconomic forces may come into play which work against such occurrences.A complete list of absolute ceilings and floors for economic fluctuations

in our Keynesian monetary growth model could be the following:

• V�V��

for the rate of employment,• U�U

��for the rate of capacity utilization,

• � � 0 for inventory holdings,• I�� K for net investment (gross investment I� �K� 0),• r� 0 for the nominal rate of interest,• � �x for real wages.


The first two items state that there are two constraints for the output offirms at each point in time t, one, (Y�

��U

��y�K), determined by the size

of the capital stock which is in existence in t and which describes themaximum usage to which the physical means of production can be put(y�K the normal usage), and one, (Y�

��V

��xL), which describes the

maximum of the labor effort available from a given labor force L (xL thenormal usage) at each point in time. The output that is actually producedat each moment of time is thus given by

Y�min�Y�I,Y��,Y�

��. (6.78)�

This equation should be used in the place of equation (6.78) when suchlimits are approached.It can, however, be expected that the behavior of the economy changes

significantly before such limits are reached. Furthermore, the value V��

will be considerably larger than the value 1 of normal labor force utiliz-ation, since it is meant to represent absolute full employment, which maybe considered to be more than twice as high than the NAIRU level 1 ofnormal employment, due to the many labor time reserves that exist withinthe firm as well as outside of it. Finally, the other limit U

��is not so

absolute as it may appear in the present context. In view of chapter 5 andits treatment of neoclassical factor substitution this limit is given by themarginal productivity relationship which, however, need not be a bindingconstraint if, for example, firms decide to sell additional production at aprice less than marginal wage costs in order to defend their market shares.The marginal productivity condition may thus be important for pricingand investment considerations, as we have used it in chapter 5, but not asan absolute limit to production as it is used in non-Walrasian regimeswitching analysis.The third item in the above list states that inventories cannot become

negative. It is also not so binding as appears at first sight since unfilledorders can be, and in fact are, treated as negative inventories in the presentmodel. These are subsequently served on a first-come first-served basisuntil inventories become positive again.The fourth item, namely, that gross investment remains non-negative,

again is not as binding as appears at first sight. This is so since thedepreciation rate may become endogenous in times of crisis where grossinvestment approaches zero.These items are all quantity constraints, while the last two items on the

above list represent price or value constraints. Negative nominal rates ofinterest r will not come about due to the behavior of asset markets if thisfloor is approached (which, however, demands the appropriate introduc-tion of a nonlinear money demand function and the like). Finally, the


mechanism that keeps real wages � below labor productivity x is not soobvious and has been controversial throughout the history of economictheory. In the presence of smooth factor substitution, see again chapter 5,one may, however, argue that prices going below marginal wage costs willspeed up the price level dynamics by so much that the occurrence ofnegative profits is generally prevented.Of course, prices p and w as well as the capital stock K have to stay

positive also, but this is assured by the formulation of their dynamics interms of rates of growth. As should be obvious from the above arguments,the barriers just listed, when approached, demand the integration of vari-ous types of nonlinearities (or additional reaction patterns such as over-time work, changes in the participation rate and immigration in the case ofthe full employment barrier) that may often prevent the described boundfrom actually being reached. The Keynesian effective demand regime maytherefore be considered as the generally prevailing one, though it cannot betotally excluded that the capital stock or the situation on the labor market(outside and inside of firms) may cause a departure from this regime incertain extreme situations.Astonishingly, however, all of the above additions to our demand con-

strained Keynesian model of monetary growth can be bypassed in manycircumstances when one simple fact of modern economies is taken intoaccount and added to the model, namely, the nonexistence of an economy-wide wage deflation w� 0. In an inflationary economy, workers maydemand very small nominal wage increases in the face of high unemploy-ment, i.e., they may not attempt to resist real wage decreases when theyoccur in this way. By contrast, the resistance to nominal wage decreasesmay be formidable due to the institutional structure of the economy. Suchand further related arguments have been put forth in a forceful way byKeynes (1936)�� in particular, and they here provide the basis for a simplemodification of the money wage Phillips curve (6.81) we have employed sofar, w�min��

�(V� 1)�

�p� (1�

�)�,0�.

This modified wage equation, which excludes the occurrence of a nom-inal wage deflation, has dramatic consequences for the stability and thepattern of fluctuations that are generated by the thereby revised model.This will be demonstrated here by a series of simulations of this extended

�� Laxton, Rose, and Tambakis (1997) have recently continued the discussion on the type ofnonlinearity that may characterize the Phillips curve, but then focus immediately on theconventional type of a single Phillips curve that is viewed to summarize labor and goodsmarket nominal adjustment processes. They argue in particular that the curvature of thePhillips curve is of decisive importance for the success of stabilization policies. Periods ofpersistent inflation (as they reappear periodically in reality and in the model that isinvestigated below) and policy issues are discussed from a broad perspective in Cagan(1979).


model which will create economically meaningful trajectories for all rele-vant variables despite pronounced increases in adjustment parameterswhich formerly rapidly led to purely explosive situations.Let us briefly describe how the model of the preceding subsections is

modified by the above reformulation of the money-wage Phillips curve.The wage and price adjustment equations of those subsections can berepresented in the form

w� ��(V� 1)�

�(p��),

p�� (U� 1)�

�(w��),

which gives rise to the expressions

w� �� [��(V� 1)�

��(U� 1)], (6.134)

p�� [ ��(V� 1)� �

�(U� 1)]. (6.135)

The simultaneous determination of wage and price deflation is therebysolved and shows that both inflation rates depend on the state of excessdemand in the market both for labor and for goods and on expectedmedium-run inflation. Subtracting the second from the first equation thengives the law of motion of the real wage we have employed so far. Yet, whenthe rule of downwardly rigid nominal wages applies, i.e., in the case where

[��(V� 1)�

��(U� 1)]� �� 0 (6.136)

holds true, we have

w� 0, p��(U� 1)� (1�

�)�,

and thus get for the real wage dynamics in this case

�� (U� 1)� (1�

�)�. (6.137)

This is the modification to be made to (6.119) whenever the inequality(6.136) holds true. Furthermore, both (6.121) and (6.122) make use of theexpression

p�� [ ��(V� 1)� �

�(U� 1)],

which in the case of the above inequality must be replaced by

p�� (U� 1)�

��. (6.138)

This completes the set of changes induced by the assumption of downward-ly rigid nominal wages.Let us now look at the consequences of this simple modification of the

model. A first example is provided by figure 6.8. This figure is based on thedata of figure 6.7 (

�� n, i.e., no steady state inflation in particular, and


Figure 6.8 No steady state inflation (�� n� 0.05, �

�� 0.292 as in figure 6.7)

also h�� 0.08,�� 0.4), and differs from the model of that figure only by

the above extension of the Phillips curve. In this case the revision of themodel has two basic consequences:

• The steady state of the model is now (for �� n) no longer uniquely

determined in the interior of the phase space as far as rates of employ-ment V

�(and l

�) are concerned. The rate V

�may now be lower than ‘‘1’’

in the steady state, since the then implied wage deflation is prevented bythe above change in the wage adjustment mechanism of the model (allother steady state values are the same as before).

• The set of steady states of the revised model is now globally asymptoti-cally stable in a very strong way (see figure 6.8 for an example). Due tothe changed behavior of workers the economy is rapidly trapped in anunderemployment equilibrium that may be much higher than theNAIRU rate of unemployment of the former steady-state situation.

We thus have that downward wage rigidity prevents the fluctuationsshown in figure 6.7 in quite a radical way, but that this is generallyaccompanied by a more depressed labor market in the steady state thanpreviously. These results are in our view due to the fact that there is a flooror ratchet built into the model right at the edge of the steady state.


Figure 6.9 Steady-state inflation (�� 0.1� n� 0.05) and period 1 limit cycles

(�� 2)

Figure 6.10 Steady-state inflation and period 4 limit cycles (�� 10)


Figure 6.11 Steady-state inflation and period 16 limit cycles (�� 11)

Figure 6.12 Steady-state inflation and complex dynamics (�� 26)


This observation suggests that there will be more fluctuations if there issteady state inflation, i.e., if

�� n is assumed, since the behavior of the

economy is then only modified further away from the steady state, which inthis case is again uniquely determined as in the preceding model. Locally,the model is thus of the same form as that of the preceding subsection. Theinteresting question then is whether the dynamical behavior is again rad-ically modified by the ratchet-effect situation that the level of nominalwages may rise, but cannot fall. Figures 6.9—6.12 illustrate the dynamicalbehavior for a wage adjustment speed �

�that varies from 2 to 26, i.e., over a

range where the previous model would have collapsed immediately.This series of figures indicates the existence of a period-doubling route to

complex dynamics for the systemwith a kinkedmoney-wage Phillips curvefor extremely high adjustment speeds �

�of nominal wages w, with ampli-

tudes of fluctuations that stay within economically meaningful bounds.Note however that wage inflation can be as high as 130 percent during thissequence, and that inventories may become slightly negative in the lastfigure 6.12 where the case �

�� 26 is considered.

Figures 6.9—6.12 each show the three projections of the six-dimensionaldynamics onto the� � l, them�� and the y� � subspaces as well as thedevelopment of wage inflation as a time series. This series of figuresdemonstrates several things about the behavior of the model. First, themodel is now extremely viable, but, as expected, no longer asymptoticallystable. Second, the model exhibits large but economically meaningfulpersistent fluctuations. Third, the model undergoes a period-doublingsequence as the parameter �

�is increased. Fourth, the model shows only

weak changes in amplitude while the parameter ��is increased significant-

ly. Fifth, the economic length of the cycle stays approximately 20 years,while the mathematical period of course doubles along the period-doubl-ing route; Sixth, the dynamics eventually becomes complex for parameter��between 11 and 26. The dynamics of the naturally nonlinear model is

thus radically changed from a global, though not from a local,�� perspec-tive whenwe allow for a ‘‘natural’’ kink in themoney-wagePhillips curve inconjunction with a sufficient degree of steady-state inflation.Figure 6.13 supplements the preceding figures (for the case

�� 0.1) by

showing a bifurcation diagram corresponding to the parameter ranges for��we have allowed for above. Here we seem to see period doubling and

complex behavior also at low values of ��(below 1), then a second

sequence of period doubling setting in at approximately �� 4.3 and

complex behavior emerging between ��

� 11 and �� 14.

However, we need to be cautious in interpreting this diagram as evidence

�� Where the Hopf-bifurcation analysis of the preceding sections still applies.


Figure 6.13 A bifurcation diagram for the dynamics considered in figures 6.9—6.12

Figure 6.14 The largest Liapunov exponent of the dynamics considered in figure6.13

of chaotic motion. This becomes apparent, for example, when the first ofthe above ‘‘period doubling routes to chaos’’ (�

�below 1) is investigated in

more detail. There it is in fact found that the bifurcation diagram thensolely shows the long transient behavior due to the sluggish adjustment ofmoney wages.� In figure 6.14 we therefore add to the above calculations apresentation of the largest Liapunov exponent, which provides evidencefor chaotic behavior in the range where it becomes positive� (which seems

� A similar observation does not hold for the second sequence of period doublings (��above

2), as further numerical investigations have shown.� See Parker and Chua (1989) for the details on such Liapunov exponents.


Figure 6.15 A test for sensitivity with respect to initial conditions for the above-shown attractor (�

�� 20)

to be the case for �� 11 already). Finally, the dynamic system then also

displays sensitivity to initial conditions as shown in figure 6.15 for theparameter value �

�� 20.

All these simulations suggest from their different perspectives that flex-ible wages (even if they are downwardly rigid and thus guarantee economicviability) can introduce significant turbulence into the dynamics of Key-nes—Metzler growth type, a turbulence that is accompanied by largerfluctuations in the state variables of the dynamics the larger the speed ofadjustment of money wages becomes. Sluggish money wage adjustmentsare thus preferable from the point of view of the simplicity and the size ofthe resulting business fluctuations.

6.4.4 Concluding remarks

In Keynes (1936) it is stated:

Thus it is fortunate that workers, though unconsciously, are instinctively morereasonable economists than the classical school, inasmuch as they resist reductionsof money-wages, which are seldom or never of an all-round character . . . (p.14)


The chief result of this policy (of flexible wages, CC/PF) would be to cause a greatinstability of prices, so violent perhaps as to make business calculations futile.(p.269)

Our working Keynesian model of monetary growth has demonstrated thevalidity of this view by means of numerical simulations of a system of lawsof motion of considerable completeness and complexity. These simulationshave shown that the assumption of downwardly rigid money wages, andthe kink in the Phillips curve that is implied thereby, can alter the dynamicsof the Keynes—Metzlermodel of monetary growth in a very radical fashion,making it not only mathematically but also economically viable even forvery high (upward) adjustment speeds of money wages.�� Of course, othersources of additional stability may also exist, but must be left for futureresearch here.

6.5 Outlook: macroeconometric model building

We have presented in section 6.3 our working model of Keynesian monet-ary growth as a synthesis of the Keynes—Wicksell model type and our basicKeynesian monetary growth model of chapter 4, by way of systematicextensions of these two model types. We have furthermore shown inchapter 5 how smooth factor substitution can be introduced into thismodel type and in chapter 4 how the model can be extended to includeHarrod neutral technological change, wage taxation, p-star expectations,and a more refined interaction of expectations and actual inflation in theform of a wage—price spiral. Taken together, we have thus arrived from thetheoretical point of view at a fairly developed presentation of Keynesianmonetary growth dynamics by way of a systematic extension of the earlyKeynes—Wicksell approach to the description of the interaction of labor-and goods-market disequilibrium in a monetary economy.Powell and Murphy (1997) have recently published a detailed presenta-

tion of the so-called Murphy model for the Australian economy thatattempts to show the theoretical foundations of their modern macro-econometric modeling of a small open economy. They use in their guide tothis model the version of it which consists of 100 scalar equations including24 behavioral equations and, of course, many accounting identities. In thecourse of writing the present book, however, it became more and moreapparent to us that there is a close relationship between the model typeconsidered in Powell and Murphy (1997) and the one at which we havenow arrived.Of course, Powell andMurphy have to consider an open economy right

�� Which as we have seen introduce considerable turbulence into the considered dynamics.


from the start, whereas we have restricted our attention in our hierarchy ofmacrodynamic model building to closed economies throughout. There is,however, work in progress�� where we extend our model types to includeinternational trade in goods and financial assets along lines suggested bythe seminal paper of Dornbusch (1976) on overshooting exchange ratesand the many extensions of it that now exist in the literature. The Dorn-busch model is also central to the Murphy model, where basically athree-goods approach is adopted (one traded and one nontraded commod-ity for the Australian economy and one international good) in conjunctionwith an asset-markets approach with rational expectations, as in manymore recent presentations of the Dornbusch model. For a detailed dis-cussion of the open economy aspects of the Murphy macroeconometricmodel, as well as of our theoretical working model for open economies andfurther extensions of it, we refer the reader to Chiarella et al. (1998) andChiarella and Flaschel (1998c,d,e,f), and will concentrate in these briefremarks on the other structural equations of the two approaches to becompared.With respect to the remaining equations of the Murphy model, which in

number are not many more than the equations we have used to describethe Keynes—Metzler model in section 6.3, there exist in particular thefollowing set of relationships between our theoretical approach and theapplied one of Murphy. We stress here that the following list of theserelationships is, however, neither complete nor very detailed, but refer thereader again to Chiarella et al. (1998) for a thorough discussion of theMurphy model and related models and their similarities to and differencesfrom our Keynesian working model. Furthermore, we pay no attentionhere to the particular types of lag structure and behavioral nonlinearitiesthat are assumed in Powell and Murphy (1997), but compare only thequalitative features of the two model types.

• The money-wage Phillips curve of the Murphy model is of the typeconsidered in the next chapter of this book, i.e., it includes the rate ofchange of the rate of employment as an argument besides allowing alsofor the inclusion of technical change and for a purely adaptive scheme ofinflationary expectations formation.

• The price dynamics for the nontraded good, though presented in a quitedifferent format (based on marginal cost calculations), is in fact of thetype we described in chapter 5 in the presence of smooth factor substitu-tion.

• There is no forward-looking behavior in the interaction of wage andprice dynamics.

• The short run in the Murphy model is always of Keynesian type, with

�� See Chiarella and Flaschel (1998c,d,e,f ).


output being determined by actual sales and planned inventories. Thereare thus no disappointed sales expectations in theMurphy model, whichimplies that it makes use of a combination of goods-market equilibriumand Metzlerian inventory adjustment rules, a situation that somehowlies between our Keynesian model of chapter 4 and the Keynes—Metzlermodel type of the present chapter.

• Asset markets are modeled in a less restrictive way using rational expec-tations equilibria on the basis of the perfect substitute assumption. Thesemarkets are, furthermore, as in our Keynes—Metzler model, explicitlyrepresented only with respect to money-market equilibrium, in a waythat is close to our representation of the stock equilibrium in the moneymarket.

• Households in the Murphy model are only of one type and representedby a consumption function of Ando—Modigliani type.

• There is amuchmore advanced treatment of labor supply in theMurphymodel, yet no endogenous determination of natural growth or natural(un)employment, which we consider in the next chapter.

• The production of firms is based on nested CES technology assumptionswith three inputs and two outputs, and thus considerably more complexthan our two-factor/one-output neoclassical production technology ofchapter 5.

• Fixed investment behavior is basically of the same type as the one weconsider in our Keynes—Metzler approach, though Powell and Murphy(1997) represent our explicit usage of the rate of capacity utilization inthe form of medium-run competitive conditions (based on marginal costrelationships).

• Taxation and government expenditure, but not money supply, obeyrules that differ from the still simple ones we are using and which areformulated in Powell and Murphy (1997) to some extent from theperspective of getting asymptotic stability of the steady state.

• There is a unique interior steady state in the Murphy model (as in ourmodels).

• There is no regime switching as the economy departs from the steadystate, since— actual prices may be lower than the competitive ones so that firms areassumed to supply what is demanded, going beyond their aggregatesupply schedules when demanded in order to satisfy their customers(consumption and investment demand in particular) at each momentin time. There are, however, rising prices in such a situation, but neverrationed consumers or investors;

— there is always enough labor time available (additional labor that canbe recruited if necessary or through overtime work of the employedlabor force, see also our next chapter in this regard).


• As in our Keynesian approach of chapters 4, 5, and 6, the Keynesianregime of the neoKeynesian rationing approaches to short-run equilib-rium (which also consider so-called Classical regimes and regimes ofrepressed inflation) is the only one that is relevant for the description ofthe temporary positions of the economy, while the other two regimes areeither only present in the medium-run adjustment behavior of the econ-omy or are of a purely hypothetical kind (since they do not come intooperation in the way it is formulated in the neo-Keynesian literature).

This closes our brief comparison of models of theMurphy and the Keynes—Metzler type.We hope to have shown thereby that our theoretical workingmodel is in fact already fairly close to an important and modern type ofmacroeconometric model, which therefore may be approached by extend-ing further and further our hierarchical approach to models of monetarygrowth. The case of such macroeconometric models can thus be represen-ted in the compact and consistent form of our working model (with allbudget restrictions that are necessary for such consistency reasons). It canthereby be made the subject of theoretical analysis (including numericalsimulations) with respect to the various dynamic effects that are conscious-ly or unconsciously included in the interaction of its modules — the Keynes,Mundell, and Rose effects, and the like, studied extensively in this and thepreceding chapters.The Papers and Proceedings of the American Economic Review have

recently published a discussion on ‘‘Is there a core to practical macro-economics that we should all believe?’’ with contributions by Blanchard,Blinder, Eichenbaum, Solow, and Taylor (all 1997).� Our reading of thisdiscussion is that the working model of Keynes—Metzler type we havedeveloped and analyzed in this chapter (andwhich we extend further on thesupply side in the next) provides a general prototype of an integratedmacrodynamics of the short, medium, and long run of monetary economiesthat is in many respects closely related to this discussion, also with respectto the term ‘‘practical’’, and that can at the least be used as a point ofdeparture for the further discussion of integrated macrodynamics withdemand and supply side features and their relationships with structuralmacroeconometric model building (see also the discussion on macro-economic modeling in a changing world in the introduction to Allen andHall 1997). We thus believe that we have reached a stage where traditional,but integrated Keynesian macrodynamics (with long-run supply side fea-tures) we have presented and investigated in this chapter will convincemany (but of course not all) readers with interest in such macrodynamics

� An alternative point of view is presented in the earlier contribution to such a discussion byMankiw (1990).


that such modeling can be very useful for evaluating the real progress thathas been made in the search for reliable structural macromodels of theshort, medium, and long run in the recent past. Finally, there is work inprogress (see Chiarella and Flaschel 1998f, Chiarella et al. (1998, 1999)where further attempts are made to synthesize to some extent (or at least tocompare) the working model of this chapter with recent contributions tomacrodynamics from various schools of economic thought, to integrateinternational trade in goods and financial assets into it, and to embed allthese achievements into a theoretical discussion of important macro-econometric structural model building.


7 The road ahead

In this chapter we shall provide one final important example of how the‘‘proper’’ Keynesian prototype model of monetary growth of this book, theworking model of the preceding chapter, can be further improved and cangive rise to considerably more refined adjustment processes on the macrolevel, here specifically with respect to the labor market (outside and insidethe firm), long-run employment and long-run growth. We shall then closethis chapter with a brief summary of what we have achieved in this bookand what has still to be done by enumerating the weaknesses, gaps andshortcomings that remain in our modeling of a Keynesian monetarygrowth model. As will be obvious from this list, it is not possible to removethese shortcomings of our working model in a single book, or even in onefurther book.Instead, we will concentrate in this final chapter on one important

problematic characteristic which our working model still exhibits concern-ing the exogeneity of the full employment rate of employment V� and thenatural rate of growth n of the economy. Furthermore, we refine theadjustment processes on the labor market and endogenize the trend com-ponent in the investment function, thereby in sum allowing for an en-dogenous determination of the steady-state rate of growth as well as thesteady-state rate of (un)employment.One consequence of our choice of endogenizing these rates is that there is

now hysteresis� in the dynamics, i.e., these endogenous rates are no longeruniquely determined, but now depend on historical conditions that remainpersistent in the long-run behavior of the trajectories of the dynamics. Also,in order to allow for an adjustment of the rate of employment, we have to

� By ‘‘hysteresis’’ we refer to the phenomenon whereby a dynamical system may have acontinuum of steady states so that the attractor to which the economy converges isdependent upon the initial conditions of the trajectories. Hence the dynamics of the systemwill exhibit path dependency (see Franz 1990, ch. 1, for a discussion of hysteresis ineconomics).

340

distinguish now between the rate of employment that, on the one hand,refers to the labor market and, on the other hand, is the one within firms(the rate of employment of the employed labor force). In this way weintroduce further delays in the adjustment of the rate of employment on thelabor market (by distinguishing inside from outside effects), thereby in-creasing the realism of our description of the employment relationships.We therefore here reduce our reliance on the use of so-called natural ratesin the formulation of our Keynesian monetary growth dynamics and takeaccount of further feedback loops in labor market adjustment processes.However, instead of a full description of goods-market disequilibrium

and its dynamic consequences, as in our Keynes—Metzler working model,we consider in this chapter again only the ‘‘naive’’ version of the goods-market disequilibrium adjustment process, i.e., the simple dynamic multi-plier story of the KT model of section 6.2 of the preceding chapter. Thisversion of the Keynesian model with both sluggish price and quantityadjustments is mathematically simpler to treat and, though somewhatunconvincing from the viewpoint of economic consistency, not unrelatedto the mathematical structure of the monetary growth model with a fulldescription of goods-market adjustment processes as we have seen inchapter 6. This output adjustment process will be supplemented here by alabor force adjustment process of a similar type, which in sum gives rise to acore dynamics of the model that is of dimension six, as the Keynes—Metzlermodel with its full description of the inventory adjustment process. In sum,this chapter thereby achieves the integration of sluggish quantity adjust-ments of output and employment with endogenous trend growth as well aswith an endogenous determination of the long-run rate of employment ofthe labor force in the Keynesian monetary growth framework introducedin chapter 4 as theKeynesian extension of theKeynes—Wicksell frameworkof the sixties and early seventies.

7.1 Endogenous long-run growth and employment

One purpose of the monetary growth model of this chapter is to justify theassumption � n(� const.) which has been employed throughout thebook as a description of the trend component in investment behavior andhas been interpreted as a very crude expression for the ‘‘animal spirits’’component of this behavior. This section now endogenizes the rate , still ina very simple way, by means of a certain self-reference that is here assumedto characterize the investment plans of firms on the macro level. Thisapproach to an explanation of the generally given rate (� n� const.)represents the simplest way available for its endogenization. One implica-tion of this new assumption will be that, since the model need not be locally

341The road ahead

asymptotically stable, the rate need not converge to the steady-state raten of natural growth. The earlier assumption of � n(� const.) need there-fore be fulfilled only ‘‘on average.’’This average, however, will also be treated as endogenously determined

in the following, since it is not very plausible that labor supply follows agiven trend in reference to which the steady state can then be determined.Trend growth, including labor supply growth, must be explained by theoryand not simply be assumed as given for the explanation of cycles around it.The long-run growth path cannot be determined by ‘‘closing’’ the systemfrom the side of labor supply as far as the evolution of industrializedeconomies is concerned.Finally, the so-called ‘‘natural’’ rate of (un)employment is also not a

datum brought about by natural forces, but is a consequence of the processof capital accumulation and its impact on the labor market. A model ofmarodynamics must therefore sooner or later offer an endogenous expla-nation of this NAIRU-based rate of employmentV� . Again this will be donein a particularly simple way here.�For recent discussions on the NAIRUof the ‘‘Natural Rate Hypothesis,’’

which, however, generally differ significantly from the approach that ischosen in this chapter, the reader is in particular referred to the volumeedited by Cross (1995), to the papers by Phelps and Zoega (1997), Saint-Paul (1997), and Murphy and Topel (1997), to Allen and Nixon on ‘‘Twoconcepts of the NAIRU’’ in Allen and Hall (1997), to Summers (1990), toFair (1997a,b), and to the discussion published in the Journal of EconomicPerspectives with contributions by Blanchard and Katz, Galbraith, Gor-don, Rogerson, Straiger et al., and Stiglitz (all 1997). Our point of view inthe present chapter is that we use a very simple endogenization of theNAIRU, the nonaccelerating inflation rate of (labor) utilization, alongsideother adjustment processes in the labor market, in order to discuss the roleof this rate in a high order integratedmacrodynamics and to get first resultson the endogenous determination of the long-run steady and nonsteadyevolution of closed economies. Of course, further work which integratespart of the approaches of the papers just mentioned is needed here to makefurther progress on the endogeneity of long-run growth as it derives fromadjustment processes in the markets for labor, and of the trend growth rateof the capital stock.Viewed from the perspective of what has been shown so far in this book,

it would be best to choose the Keynes—Metzler model of section 6.3 andinclude the above-proposed modifications into it, though it may be prob-

� Our approach parallels in method, but not in substance, the approach chosen in van deKlundert and van Schaik (1990) for the evolution of inflation and the capital stock in aKeynesian environment.


lematic to do the same for its extreme equilibrium limit case considered atlength in chapter 4 (as the ‘‘textbookmodel’’ of IS—LMgrowth). Section 6.4has, however, shown that it may be legitimate to make use of an intermedi-ate case, theKTmodel of section 6.2, in order to present and investigate theabove-proposed endogenization of ‘‘natural rates,’’ since the Keynes—Met-zler growth model can be reduced to a model of KT type if some mildalterations in its structural equations are accepted. We therefore here takethe KTmodel type of section 6.2 as starting point for our description of anendogenous determination of the trend component in investment behavioras well as in labor supply and of the NAIRU-based rate of employment.With respect to this market, we furthermore will distinguish now betweenthe employment of the employed and the rate of employment of the laborforce in order to separate the immediate impact of a fluctuating demand forgoods on labor effort from its medium-run consequences for the pool, orthe reserve army, of the unemployed. This adds some further inertia to themodel by making the employment decisions of firms a more indirect one.This section thus offers an insider—outsider approach to the labor marketwith endogenous solutions for the trend growth rate in this market as wellas for the economy as a whole, plus a historically determined endogenouslong-run level of the employment rate.

The equations of this version of the KT model of monetary growth ofsection 6.2 read in the case of a given tax ratio t� (see below):


��w/p, u��/x,� � (Y� �K� �L)/K, (7.1)


%� 1. (7.2)



�pY� h

�pK(1� �)(r� � r), (7.3)

C��L� (1� s�)[�K� rB/p�T ], s

�� 0, (7.4)

S��L�Y�


� s�[�K� rB/p�T ]� s

�Y��

� (M� �B� � pE� )/p, (7.5)

L1 � n, n� � ��(n(V, )� n), n(V, )� n

�(V�V)� n�(� )� n.(7.6)


See Lindbeck and Snower (1988) for a collection of articles on the insider/outsider issue. See also Chiarella and Flaschel (1998b) for a related presentation of this final model of ourbook and its theoretical and numerical implications.

343The road ahead


L�Y/x,x� const.,V�L�/L,V��L/L�, (7.8)

L� �� L�� (L�L�), (7.9)

I� i�(�� (r� �))K� i

�(U�U� )K� K, (7.10)

pE� /p� I� (S� I)�Y� �K�C�G�Y�Y� I, (7.11)

K1 � I/K�S/K, (7.12)

� ��(K1 � ). (7.13)


t�� (T� rB/p)/K� const., (7.14)

G�T� rB/p��M/p, (7.15)


M1 ��, (7.17)

B� � pG� rB� pT�M� [� (��

�)M]. (7.18)



�pK(1� �)(r� � r)[B�B,E�E], (7.19)

pE� (1� �)�pK/((1� �)r��), (7.20)

M� �M� ,B� �B� [E� �E� ]. (7.21)

6 Disequilibrium situation (goods market adjustment):

S� pE� � S

��S

��Y� �K�C�G� p

E� � I, (7.22)

Y�C� I� �K�G, (7.23)

Y1 � ��(Y/Y� 1)� ��

�((I� S)/Y), (7.24)

N� � ��K� S� I,S�S

�� S

��Y� �K�C�G. (7.25)

7 The dynamics of the labor market NAIRU:

V�� v� (V�V� ). (7.26)


w� ��(V�V� )��

��(V�� 1)�

�p� (1�

�)�, (7.27)

p��(U�U� )�

�w� (1�

�)�, (7.28)

�� (p��)��(�� n� �). (7.29)


With this model we have returned to the I� S—LM KT model of section6.2 (see (6.20), p. 285) in order to formulate and investigate another exten-sion of it (besides the Keynes—Metzler model type), one that now dispenseswith the exogeneity of most of its ‘‘natural’’ rates. In addition to that, weshall now also make use of an extended labor market Phillips curve which,as in insider—outsider approaches of the literature, distinguishes betweenthe employment rate V of the labor force and the employment or utiliz-ation rate V� of the employed (the insiders). This distinction is of import-ance since it now takes account of the fact that there is in general only asluggish response of the rate of employment to the actual employment thatexists within firms. The latter is governed by the fluctuating state ofeffective demand on the market for goods, which is first of all met by firmsthrough a changing rate of employment of the employed labor force andonly with a time delay through a change in the number of workersemployed.Such a delay in the response of firms to changes in effective demand with

regard to their hiring or firing of workers may give additional force to ourselection of the Keynesian demand constrained regime as the general one,and our neglect of the regime of so-called repressed inflation (where thelabor supply has become an absolute constraint on the output of firms) andthe so-called Classical regime (where the capital stock takes on the role ofsuch an absolute constraint, a state of absolute capital shortage). Theformer regime will be avoided here by this increased flexibility in theemployment decision of firms and by means of the reactions of wages tothis insider—outsider characterization of the labor market, plus also theresulting profitability effects on the investment plans of firms. Capitalshortage U� 1, however, can here only be avoided through timely reac-tions in the investment decision of firms, possibly speeded up by a rate ofprofit which then significantly increases relative to the real rate of interest.As we have shown in the preceding section, the inventory policy of firmsmay be an additional argument for the exclusion of the Classical regime ina certain neighborhood of the steady state.Nevertheless, regime switching may occur under certain circumstances

and has therefore to be integrated in some way or another into suchKeynesian demand driven systems at a later stage. At that later stage it willalso become necessary to provide an endogenous explanation of the secondtype of NAIRU-based rate of employment we have employed in thisKeynesian model, the desired or normal rate of capacity utilizationU� � 1of the capital stock. This, however, will also demand a more elaboratedescription of the decisions made within firms.In the present context we content ourselves with an endogenous expla-

nation of the other natural rates of themodel. Themain reason for this final

345The road ahead

modification of our Keynesian model is that we want to indicate to thereader that the assumption � n we have employed so far can indeed bejustified to some extent, but that it is not possible in a truly general growthmodel to state that it is ‘‘closed,’’ and thus made a determinate modelthrough a given growth rate of the labor force or an independent invest-ment demand function or the like.� In the above model these ‘‘data’’ ofcertain steady-state analyses are determined and explained simultaneouslyso that none of them can be given in advance in order to derive steady-statepositions from it. In fact the choice of the steady state of the above modelwill no longer be unique here, since it turns out that there is now hysteresisin the evolution of the economy, and thus dependence of the steady state inthe long run on the specific type of evolution that has taken place or willtake place in response to exogenous shocks. The ‘‘natural’’ rates of growthof both the capital stock and the labor force n (which are generated bydifferent sectors of our economy with quite different objectives in mind)will here in general converge to (or cycle around) each other and, ifconvergent, converge to a value which is not predetermined by the par-ameters of the model.Since we have returned to the assumption of the KTmodel, we can again

use actual output in the calculation of the rate of profit � in equation (7.1) inplace of the expected demand concept we used in section 6.3. Furthermore,the explicit treatment of inventories is again no longer compelling, sinceinventories do not feed back into the other dynamic equations in thepresent formulation of the model (and will thus be suppressed here).Output adjusts again in view of the discrepancy that exists between itscurrent level and actual aggregate demand, and also in view of the fact thatthe economy is assumed (by firms) to grow on an average with the rate .In the household sector we now assume in (7.6) that the time rate of

change of the growth rate of the labor supply depends positively on thediscrepancy between a target value for it, n(V, ), and the current value n ofthe rate of natural growth. This target value n(V, ) in turn dependspositively on the current state of the labor market as measured by the rateV and also positively on the currently existing growth climate as measuredby the term . It is here assumed that there are segments in the economywhere people are just waiting (to be called) to enter the labor market in thecase of brighter prospects, which are here represented by these two vari-ables. We have assumed this particular approach to changes in the rate ofnatural growthmainly because of two important subcases, namely, �

��

(i.e., n� n(V, )), and �� , n(V, )� (i.e., n� � �

�(� n)).

In the first case the variable n is determined as a simple statically

� See Marglin (1984a,b) with respect to this type of approach.


endogenous variable with no need of a further dynamic law for its evolu-tion, while in the second case we simply assume that it adjusts to thelong-run rate of growth that firms expect to come about.There are two alterations with respect to the assumed behavior of firms.

One is that we now distinguish between the rate of employment V of thelabor force (on the labor market) and the employment rate V� of theworkforce employed by the firms. This latter rate is determined by thetemporary state of goods demand, which is known to firms, and by theexact basis of their output decision (which in turn determine the employ-ment within firms). Equation (7.9) describes the adjustment of the work-force L� that is chosen by firms in the light of their evaluation of theemployment of their labor force and in the light of the trend rate of growththey expect to come about. The other change in the description of firms isthe assumption (7.13), which states that the investment climate they jointlygenerate will feed back on their views of the trend rate of growth in apositive fashion.The description of the government sector and of the asset markets is the

same as before and the adjustments in the output decisions of firms in viewof current aggregate demand are described as in the KT model of section6.2.For the NAIRU-based rate of employment V� , we assume in equation

(7.26) that it responds to the state of the labor market. An employment rateabove (below) V� feeds back positively (negatively) on the rate V� by enlarg-ing or reducing that segment of the labor market where labor can beconsidered as ‘‘trained,’’ i.e., sufficiently experienced in the use of the giventechnology and not disqualified through a longer period where it has beenunemployed. We do not go into the details of such a hysteresis-creatingloop between the actual and the long-run rate of employment, since weonly want to employ this simplest hypothesis conceivable for describingsuch an effect (see Cross 1987 for a more elaborate treatment of such effectsand for notes on the literature, and Heap 1980 for an early approach, andits justification, that is closely related to the one we have employed here).Our final equation concerns the money-wage Phillips curve (7.27) which

describes the outside and the inside effect of employment as we havealready sketched it above. Deviations of the labor-market employmentrate from the NAIRU-based rate V� lead to corresponding effects on therate of change of money wages, as do deviations of the degree of utilizationof the labor force within firms from the normal utilization rate of theemployed which is given by 1. Combined with equation (7.9), this generalform of a money-wage Phillips curve marries in particular the views ofPhillips (1958) and Kuh (1967) which therefore are complementary to eachother, but not alternatives, as Kuh believed. Inserting (7.9), appropriately

347The road ahead

transformed, into (7.27) implies that the rate of employment enters thePhillips curve in proportional (V) as well as in derivative form (V1 ), leavingas empirical questions the extent to which each of these terms determinesthe rate of change of money wages. Note here that the derivative termmustbe integrated in order to give rise to the level formulation used by Kuh(1967). Neglecting the micro—macro distinction with respect to the just-considered Kuh-component of the Phillips curve, the integrated derivativecontrol term in it can also be related to the so-called wage curve asintroduced and investigated extensively in Blanchflower and Oswald(1990, 1994, 1995).We stress once again that each building block in the model may be

subject to considerable change, and that it is primarily the completeness ofthe presentation of macroeconomic interactions that is considered here asimportant. Expressions have therefore often been chosen in as simple a wayas possible in order to sketch all basic adjustment mechanisms in a Key-nesian model of monetary growth. Nevertheless, we have now obtained avery general model type for the analysis of Keynesian monetary growthwith both under- and overutilization of labor and capital.Finally, one could have called this particular extension of the KT model

aKeynes—Marxmodel. Such a denomination is intended to express the factthat we have now laid more stress on the labor market and its structuralcharacteristics than is generally customary inKeynesianmodels of the longrun. It was indeedMarx (1954, ch. 25) who first pointed, in a way that is stillrelevant, to the crucial role played by the labor market and its structure inthe shaping of the process of capital accumulation. Many of his character-izations of its structure are still relevant, although, of course, crudelyformulated from today’s perspective. This closes the description and themotivation of our above final model type of Keynesian monetary growth.

7.2 The dynamic structure of the model

The model of the preceding subsection is again easily reduced now to anine-dimensional autonomous dynamical system in the state variablesu��/x, l,m, �, y,V,V� , , and n, with an appended b� dynamics (not shown).These dynamics will be investigated in the remainder of this chapter withrespect to the new phenomena to which they give rise.� The intensive formnine-dimensional dynamical system is given by

u� [(1� �)(�

��(V�V� )��

��(y/(xVl)� 1))

� Note that the derivation of the dynamic law V1 � (� n)��(V�� 1) is easily obtained

from the expression L1 �� (V�� 1), which in turn is an obvious consequence of the

lawofmotionwe have assumed forL�.Note also that the law forL1 � is of the same type as theoutput adjustment rule of the KT model.


� ( �� 1)�

�(y/y�� 1)], (7.30)

l1 � n� (� i�(�� r� �)� i

�(y/y�� 1)), (7.31)

m� �� n�� l1 � [�

�(y/y�� 1)�

�(��(V�V� )

��(y/(xVl)� 1))], (7.32)

�� [��(y/y�� 1)� �(��(V�V� )� �

��(y/(xVl)� 1))]

� ��(�� n� �), (7.33)

y� �� (i�(�� r��)� i

�(y/y�� 1))y��

�(y� y), (7.34)

V1 � (� n)��(y/(Vxl)� 1), (7.35)

V�� v� (V�V� ), (7.36)

� ��(i�(� � r��)� i�(y/y�� 1)), (7.37)

n� ��(n(V, )� n), (7.38)

where we have employed the abbreviations (r� � r�):

�� y� ��l� y(1� u)� �, l�L/K� y/x (y not const.!),V��L/L�� l/(Vl)� y/(xVl),U� y/y�,r� r

�� (h

�y�m)/h

�,

g� t��m,

y� uy� (1� s�)(� � t�)� i

�(�� r��)� i

�(U�U� )

� n� �� g,i( · )� I/K� i

�(�� r��)� i

�(y/y��U� )� ,

where u��/x is the share of wages in gross national income.Assume for all following considerations that the equation n(V� ,

�)�

�has a unique positive solution

�for each meaningful level of the NAIRU

rate of employment V� � (0, 1). For any choice of this NAIRU-based rate ofemployment V� there is then a unique interior steady-state solution for theremaining state variables of the above dynamical system if one adds to itthe equation n

��

�(with

�as determined above), i.e.:

y�� y

�� y�, l

�� y

�/x, (7.39)

m�� h

�y�, (7.40)

��

�� n

�, (7.41)

��y�� t�� (n

��

�m

�)/s�

l�

, (7.42)

�� y

��

�l�, (7.43)

In the special case n(V, )� , to be investigated numerically later on, the set of steady statesis given by a surface in R�.

349The road ahead

r��

��

�� n

�(7.44)

V��V� , l

�� l

�/V� . (7.45)

The set of economically meaningful steady states of the considered eight-dimensional dynamics is thus given by a curve inR�. Themodel is assumedto have sufficiently large buffers 1�V� , 1�U� at its steady state so that theKeynesian demand regime we have assumed above to prevail at each pointin time (in each short run of the model) can indeed be a maintained alongthe trajectories of the dynamics.We stress that we have chosen all behavioral equations as linearly as

possible in order to concentrate on the ‘‘natural’’ or intrinsic nonlinearitiesof the dynamics and their implications. This in particular holds for thefunction n(V, ), which implies that there is a unique solution

�to the

equation

n(V� , �)�

�:

��n�(V� �V)� n� n�

1� n�, n�� 1

for each meaningful level of the NAIRU rate of employment V� � (0, 1).

7.3 Analysis of the employment subdynamics

In this section, we shall consider with increasing generality the special caseof the model of the previous section where the rates �n are still givenexogenously (�� 0), but where there is a delayed adjustment oflabor according to the work-time within firms (1/�

�� 0) and where the

‘‘natural’’ rate of employment follows the rate of actual employment withsome time delay (�v� � 0). This leads us to a seven-dimensional systemwhose stability properties are investigated by starting from known resultson the five-dimensional core situation (�

�� v� � 0), extending it to six-

dimensional dynamics via �� 0, and then to the seven-dimensional dy-

namics via �v� � 0. In this way we get some insights into the dynamicworkings of the model as far as medium-run aspects of the labor marketand the employment decisions of firms are concerned. We then consider inthe next section the seven- and eight-dimensional subdynamics that comeabout when the endogeneity of the two rates n and is discussed inisolation from the aspects considered in the present section. In this way weobtain partial insights into the driving dynamic mechanisms of the fullnine-dimensional dynamics of our general KT model.Let us thus first consider the subcase of the abovemodel that is described

by

�� 0, (0)� n� const.,


�� 0, n(0)� n� const.,

�v� � 0,V� (0)�V� � const. � (0, 1).

In the resulting six-dimensional subcase we therefore still assume that allnatural rates are given exogenously, so that the only extension with respectto the KT model is the treatment of the variableV, which in the KT modelwas determined as a function of the state variables of the model and whichhas now itself become a state variable that is following the development ofthe variable V�� y/(Vxl) with a time delay according to the law V� ��(y/(xl)�V).

Proposition 7.1: Assume �� 0 and V given by its steady-state

value. Then the steady state of the five-dimensional dynamical system(7.30)—(7.34) is locally asymptotically stable if the parameters h

�, �

��, �

��,

��, and �� are chosen sufficiently small and the parameter �

�sufficiently

large.

Proof: The dynamics (7.30)—(7.34) is (formally) of the type we havediscussed as the KT model in section 6.2 of chapter 6. This implies theassertion, and in particular that the economically meaningful steady stateof this dynamics is uniquely determined and of the type derived in section6.2.�

In respect of proposition 7.1, we make a number of observations. Firstly,there is also a close economic similarity between the above model (for theparameter value �

�� 0) and the KT model, since the present model then

behaves as if there were no outsiders and thus only a rationing process withrespect to insiders that leads to nominal wage reactions as they are postu-lated by the (reduced) type of Phillips curve of the KT model, now for theinsiders solely. Secondly, due to the above proposition, the stability asser-tions of section 6.2 all apply to the present limit case of the above dynamicsas well. They will therefore not be repeated here. Thirdly, the case �

�� 0

may be called the ‘‘Japanese’’ variant of the KT model, while the case�� could be considered as the ‘‘US’’ version of this dynamical system.

Proposition 7.2: Assume that the dynamic model considered in thepreceding proposition 7.1 is locally asymptotically stable with respect to itseconomically meaningful steady state. Then the steady state of the fullsix-dimensional dynamics (7.30)—(7.35) with �

��0 (given as in the KT

model and by V��V� ) is locally asymptotically stable for all parameter

values ��that are chosen sufficiently small.

Proof: Since the determinant is the product of the real parts of the

351The road ahead

eigenvalues of the underlying matrix, and since we already know that thefirst five eigenvalues have a negative real part for �

�equal to zero (and thus

by continuity also for all values of this parameter that are chosen sufficient-ly small), it suffices to show that the determinant of the Jacobian J of thefull six-dimensional dynamics (as always evaluated at the steady state) ispositive. The new sixth eigenvalue, which was zero for �

�� 0, must then

also have a negative real part (in fact it must be real in addition).Exploiting again the many repetitions of the same or of linearly depend-

ent terms in the structure of thematrix J, one can again easily show that therows of thematrix that correspond to the various differential equations canbe transformed to

u� �� const. ·Vl� �� const. ·mm� � � const. · y�� const. ·�y� �� const. · uV� � � const. · l

without any change in the determinant to be investigated. Calculating thedeterminant of this latter system then gives with respect to the signs thatare involved:

det J� (� )(� )(� )(� )(� )(� )(� )(� )� 0

which proves the proposition.�

Choosing on the other hand �� reestablishes the KT model in its

original form, since we then have V�V� always. The local asymptoticstability assertions for the five-dimensional KT model thus hold in thepresent six-dimensional version of it for small as well as for infinitely largeadjustment speeds of the outside rate of employment to the employmentneeded within firms. They are thus independent of whether the burden ofadjustment of employment to the level of intended production (followingthe state of aggregate demand with a time delay) falls on insiders or mainlyon outsiders. It is an open question here whether the observations justmade also hold for intermediate levels of the adjustment parameter �

�.

Looking on the structure of the six-dimensional system, it appears notimplausible that such an assertion can indeed be made, since the sixthdynamical law simply reappears as a component in the three other laws ofmotion of the six-dimensional case which, when it could be removed,would indeed cause the remaining dynamics to be of the KT type. Be thatas it may, the considered six-dimensional dynamics can be characterized asa very natural and basic extension of the KT model. It makes use of very


similar rules for delayed output as well as delayed labor demand adjust-ments (see the original formulations of the respective dynamical equationsin the general presentation of the model).Let us now turn to the seven-dimensional subcase of the full nine-

dimensional dynamical model (7.30)—(7.36) that allows for adjustments inthe ‘‘natural’’ rate of employment V� , i.e., to the situation:

V�� v� (V�V� ),�� 0, (0)� n� const.,�� 0, n(0)� n� const.

This extension of the previous subdynamics implies the existence of hys-teretic effects with respect to the long-run behavior of the model as the nextproposition will show.

In general terms hysteresis is a property of dynamical systems. Hysteretic systemsare path-dependent systems. The long-run solution of such a system does not onlydepend on the long-run values of the exogenous variables (as usually) but also onthe initial conditions of each state variable. These systems have a long-lastingmemory and are therefore ‘‘historical’’ systems. Loosely speaking:Where you get tois determined by how you get there. (Franz, 1990, p.2)

Proposition 7.3: Consider the seven-dimensional dynamical sys-tem (7.30)—(7.36) where �

�� 0 holds with given initial rates n� .

Then (1) for any choice of the NAIRU-based rate of employmentV� there isa unique steady-state solution for the remaining state variables as de-scribed in section 7.2. The set of economically meaningful steady states ofthe considered seven-dimensional dynamics is thus given by a ray inR ; (2)there holds det J� 0 for the Jacobian J of the considered dynamicalsystem at the steady state, i.e., at least one eigenvalue of the dynamicalsystem is equal to zero; (3) apart from the resulting path dependency withrespect to the values ofV� and l, the conditions for local asymptotic stabilitywith respect to the above ray are the same as for the previously consideredsix-dimensional dynamical system if the parameter �v� is chosen sufficientlysmall.

Proof: (1) Obvious.(2) The laws of motion for the state variables V and V� allow removal of

the expressions for the wage Phillips curve from the first four laws ofmotion as far as the calculation of determinants is concerned. It is then easyto see that both the m equation as well as the � equation can be furthersimplified and can then be shown to depend both on the state variable �solely. This makes these two equations proportional to each other and thusimplies that the determinant of the Jacobian of the dynamics must be zeroat the steady state.(3) Obvious.�

353The road ahead

7.4 Analysis of the growth subdynamics

We now turn to the two natural seven-dimensional subcases of the abovenine-dimensional dynamical model (7.30)—(7.38), each of which extends thesix-dimensional core subdynamics by the endogenization of one of its‘‘natural’’ rates of growth n and . These subcases of the general model canbe obtained by the relationships shown below. Thereafter, the eight-dimen-sional case where both rates of growth adjust simultaneously will beinvestigated.Note that we assume throughout n(V, )� in the adjustmentrule for the natural rate of growth which leaves a general consideration ofthis adjustment process for later investigations.

Subcase 1:

n� ��(� n), �v� � 0,V� (0)�V� � const.,�� 0, (0)� � const.,

Subcase 2:

� ��( · )� i�( · ),�

�� 0, n(0)� n� const.,�v� � 0,V� (0)�V�

� const.

These three extensions (i.e. the two seven-dimensional subcases and theeight-dimensional case) of the six-dimensional core dynamical system withsluggish adjustments of both output and employment will now be inves-tigated one by one.

Proposition 7.4: Consider the seven-dimensional dynamical sys-tem (7.30)—(7.35), (7.38) as described under subcase 1. Then (1) the economi-cally meaningful steady state of this dynamical system is uniquely de-termined and as described in the six-dimensional case (based on n� ); (2)the determinant of the Jacobian of the seven-dimensional dynamical sys-tem evaluated at the steady state is always negative; (3) a locally asymptoti-cally stable dynamical system in the six-dimensional case remains locallyasymptotically stable in this seven-dimensional extension of the six-dimen-sional case for all parameter values �

�.

Proof: (1) Obvious.(2) Obvious, since the new dynamical law induces only one nonzero

(negative) entry in the last row of the Jacobian to be investigated and sincethe determinant of the Jacobian to be considered in the six-dimensionalsubcase has been shown to be positive always.(3) Obvious.�

The dynamical system of subcase 1 just considered adds to the KT model


with a delayed labor-market adjustment a very simple adjustment processfor the natural rate of growth of the labor force which is simply assumed toadjust to the given trend term in the investment equation. Despite itsextreme simplicity, this extension of the six-dimensional dynamics never-theless portrays an important economic idea, namely, the view that invest-ment is the independent ‘‘equation’’ in a capitalist economy and not the‘‘equation’’ for natural growth. Or, put in another way, that labor adjuststo the conditions of capital accumulation, not capital accumulation to theconditions of natural growth (see Marx 1954, ch. 25, for such a view). Incontrast to the situation considered in the preceding proposition we do nothave hysteresis in the present extension of the core six-dimensional dy-namics. This is due to the fact that the rate of natural growth must stillconverge to a predetermined value .

Proposition 7.5: Consider the seven-dimensional dynamical sys-tem (7.30)—(7.35), (7.37) as described under subcase 2. Then (1) the economi-cally meaningful steady state of this dynamical system is uniquely de-termined and as described by the six-dimensional case (based on

�� n); (2)

the determinant of the Jacobian of the seven-dimensional dynamical sys-tem evaluated at the steady state is always negative; (3) a locally asymptoti-cally stable dynamical system in the six-dimensional case remains locallyasymptotically stable in this seven-dimensional extension of the six-dimen-sional case for all parameter values �� chosen sufficiently small.

Proof: (1) Obvious.(2) Obvious, since the right hand side of the new dynamical law can be

transformed to the form ��(n� ) as far as the calculation of determinantsof Jacobians is concerned (bymaking use of the right hand side of the law ofmotion for the state variable l). This form then implies that the seven-dimensional Jacobian must have the opposite sign to the determinant ofthe Jacobian of the six-dimensional core dynamics and thus must benegative in sign.(3) Obvious.�

The dynamical system just considered in proposition 7.5 adds to the KTmodel with a delayed labor-market adjustment a very simple self-refer-encing process for the determination of the trend rate of growth that isrelevant for long-run investment decisions. Note that in this dynamical lawfor there is no reference made to the natural rate of growth n of the laborforce. If the dynamics are locally asymptotically stable, the rate mustnevertheless approach the given rate n, due to the fact that l1� 0 impliesK1 � n, which in turn then implies � n. The process just describedmay be

355The road ahead

viewed as providing a (simple) neoclassical explanation of how these twoindependent rates of growth come together. It is useful to contrast this viewwith the also very simple Marxian view we sketched in our comments onthe preceding proposition.Of course, both viewsmust be developed furtherbefore they can be considered as sufficiently justified. Again we do not havehysteresis effects in the present extension of the core six-dimensionaldynamics. This is due to the fact that the rate must still converge to apredetermined value n. Neither subcase 1 nor subcase 2 therefore allow forhysteresis effects, as was the case for the seven-dimensional dynamics in theprevious section, which included the endogenous determination of theNAIRU-based rate of employment V� . The following proposition will,however, show that the combined adjustment processes for n and arecapable of producing hysteresis now with respect to the growth rate of theeconomy. We thereby turn to the investigation of an important eight-dimensional subcase of the full nine-dimensional dynamical model.

Proposition 7.6:Consider the eight-dimensional dynamical system(7.30)—(7.35), (7.37), (7.38) with �V� � 0. Then (1) for any choice of the trendgrowth rate in the investment function there is a unique steady-statesolution for the remaining state variables of the type described in section7.2. The set of economically meaningful steady states of the consideredeight-dimensional dynamical system is thus given by a ray in R�; (2)corresponding to the situation just described, there holds det J� 0 for theJacobian J of the considered dynamical system at the steady state, and thusone eigenvalue is always zero; (3) apart from path dependency, the condi-tions for local asymptotic stability with respect to the above described rayare the same as for the previously considered seven-dimensional dynamicsif the parameter �� is chosen sufficiently small.

Proof: (1) Obvious.(2) It is sufficient to note here that the equation governing the law of

motion of l can be expressed in the present case as a simple linear combina-tion of the two laws for n and .(3) Obvious.�

7.5 Analysis of the complete dynamical system

Finally, we come to an investigation of the full dynamical structure of theKT-model with endogenous growth and labor force participation.We hereproceed in two steps: (1) the full integration of the seven-dimensional caseswe have considered above, and (2) the investigation of the model with ageneral type of function n(V, ) as far as the dynamics of the natural rate ofgrowth is concerned.


Proposition 7.7: Consider the steady state of the full nine-dimensional dynamical system (7.30)—(7.38). Then (1) for any choice of theNAIRU-based rate V� and the trend growth rate in the investmentfunction there is a unique steady-state solution for the remaining statevariables of the type described in section 7.2. The set of economicallymeaningful steady states of the considered nine-dimensional dynamicalsystem is thus given by a surface in R�; (2) corresponding to the situationjust described, rank (J)� 7 for the Jacobian J of the considered dynamicsat the steady state, and thus two eigenvalues are always zero.

Proof: (1) Obvious.(2) It is sufficient to note here again that the equation governing the law

of motion of l can be expressed in the present case as a simple linearcombination of the two laws for n and , and that in addition the furtherlinear dependency we considered in proposition 7.3 can be shown to holdtrue in this case.�

Proposition 7.7 implies that hysteresis is now present in the model in twodimensions, concerning ‘‘natural’’ employment and ‘‘natural’’ growth. Fur-thermore, this dependence on historically given conditions of the long-runbehavior of these magnitudes does not only apply to steady states, but alsoto all other attractors that may exist for this nine-dimensional nonlineardynamical system. Since our analytical discussion has been confined tolocal stability analysis, however, it is unknown what these attractors maylook like. Clearly this would depend on the numerous parameter constella-tions this high-dimensional system allows for. At present, the attractorbehavior can only be further investigated by turning to some numericalsimulations of the model, which are performed in the next section.The above twofold hysteretic situation can again be reduced to a (single)

hysteretic evolution of the NAIRU by way of proposition 7.8.

Proposition 7.8: Consider the full nine-dimensional dynamicalsystem (7.30)—(7.38), but now given a function n(V, ) that differs from thespecial choice n(V, )� we have considered so far. Assume, furthermore,that the equation n(V� ,

�)�

�has a unique positive solution

�for (all

meaningful) V� � (0, 1). Then (1) for any choice of the NAIRU-based rate V�there is a unique steady-state solution for the remaining state variables ofthe type described in section 7.2. The set of economicallymeaningful steadystates of the considered eight-dimensional dynamical system is thus givenby a ray in R�; (2) corresponding to the situation just described rank(J)� 8 for the Jacobian J of the considered dynamical system at the steadystate, and thus one eigenvalue is always zero.

357The road ahead

With this proposition we terminate the theoretical discussion of the nine-dimensional dynamical system with endogenous growth and endogenousNAIRU (and its various subdynamics). It is obvious that we have for-mulated with these propositions only very basic results for this extension ofthe KT dynamics and its various subcases. This must suffice here as anoutlook on Keynesian monetary growth theory which attempts to en-dogenize important ‘‘natural’’ rates of economic theory.

7.6 Some numerical simulations

Before closing, we provide some numerical illustrations of the variousdynamical systems we have considered in this chapter. We here stress onceagain that the full dynamical system is characterized by a number ofadjustment lags of employment and of endogenous growth which togethermay be briefly summarized as follows:�

V�V�, �K1 ,n� (or n(V, )),V� �V.

In the following simulation studies these various routes of adjustment willbe switched on one-by-one in the order that we have discussed them above.We will thereby be provided with quantitative impressions of the workingof these various feedback mechanisms in isolation as well as in theirinteraction.Simulating our model of endogenous long-run growth and employment

on its various levels of generality (six-, seven-, eight-, and nine-dimensional)provides us with various scenarios of fluctuating growth, yet generally oneswith an increasing amplitude of the cycle.We are here back in the situation,frequently observed throughout this book, that the intrinsically deter-mined nonlinearities of our dynamic models are in general too weak togenerate overall viability of the dynamics in the case of their local instabil-ity. Once again, we therefore have to add extrinsic nonlinearities to thestructure of our models of monetary growth, and shall do this by againmaking use of the kinked money-wage Phillips curve of section 6.4.3 inorder to show some basic numerical features of the endogenous growthmodels (and their limit cases) in the presence of this extrinsic nonlinearity.Let us start with the six-dimensional core dynamical system of, in fact,

no endogenous growth, but only delayed output and employment adjust-ments. Wemake use in the following of the basic parameter set provided intable 7.1, where a number of adjustment coefficients are chosen as zero inorder to reflect this six-dimensional case properly.

� Note that we assume n(V, )� for the adjustment rule of natural growth in the followingsimulations. This leaves a general consideration of this adjustment process for laterinvestigations.


Table 7.1.

s�� 0.8, �� 0.1, y�� 1, x� 2, n� � 0.05, n

��

�� 0.05.

h�� 0.1, h

�� 0.05, i

�� 0.25, i

�� 0.5.

��

� 2.5, ��

� 2.5, �� 1,

�� 0.5,

�� 0.5.

�� 0.6, �� 1, �� v� � 0, �

�� 2, �

�� 2.

n�� 0, n�� 1, V� 1.

�� 0.08,

�� 0.08, t�� 0.08, �

��

�� 0.

We note that the initial values of the rates V� , n, and are set equal to 1,0.05 and 0.05, respectively, and that they do not change their value in thepresently considered situation. Also, in order to avoid the system gettingstuck in a depressed situation, we generally assume that there is steady-state inflation (of the kind p

��

�� n

�� 0), so that the kink in the

Phillips curve does not apply to the steady state itself (only becomingoperative at a level ‘‘somewhat below’’ it). Such a situation is represented,for example, through the simulation run of the model over a time horizonof 200 years displayed in figure 7.1.We can see in this figure that the three rates V� , , and n are still kept

constant. The figures also show irregular and asymmetric fluctuations (ofthe variable V in particular) of considerable phase length and a largeamplitude, the latter being due to the high adjustment speeds that we haveassumed here for the level of money wages, prices, and expectations, whilethe asymmetry in amplitude and time duration of depressions vs. booms isdue to the kink in the Phillips curve.� The question arises as to how thisirregular behavior will change if longer time horizons are allowed for. Anexample of this is shown in figure 7.2.Extending the time horizon to 1,000 years (and beyond) thus shows that

the irregularity of the fluctuations does not disappear from the dynamics.Some sort of complex dynamics seems to be generated in the presentsituation. This impression is confirmed by the bifurcation diagram shownin figure 7.3, which corresponds to the plots shown so far. We here see thatthe situation found to hold for the particular parameter value �

�� 2 holds

true for all values of this parameter in the range [0.5, 20]. Furthermore, theattractor that is indicated by each vertical slice in this bifurcation diagramis of a fairly fixed shape with regard to the considered parameter and itsrange.Before we leave the six-dimensional core dynamics of no endogenous

growth, let us briefly contrast the results obtained above with the situation

� Note that price level deflation is not excluded from the dynamics by a corresponding kink inthe price Phillips curve, so the kink in the figure top right is solely due to the kink inmoney-wage behavior.

359The road ahead

Figure 7.1 Phase plots and times series representations over a time horizon of 200years (6D case) (see table 7.1)

where there is no inflation in the steady state. We thus now assume��

�� 0.05 in the place of

��

�� 0.08. The consequence of this

simple alteration in our parameter set is that there is now a continuum ofsteady states below the initial one, onto which the economy can settle. Thisis due to the fact that money wages cannot fall at each of these lowersteady-state values. Though the systemmay thus be very far away from theNAIRU-based level V� � 1, it can no longer recover back to and beyondthis level, because money wages cannot fall at the new steady states of themodel. The economy (see figure 7.4) therefore gets stuck in a depressedstate, and this in a way that no longer allows for the irregular and persistentfluctuations we observed beforehand.The simulations we have thus far shown indicate that there is an import-

ant choice for monetary policy to be made between a situation where theeconomy is very stable, but maybe also (very) depressed, as in figure 7.4,and one, as in figure 7.2, where the economy in fact recovers (slowly) fromeach depression, but this at the cost of large and persistent fluctuations


Figure 7.2 Phase plots and times series representations over a time horizon of1,000 years (6D case) (see table 7.1)

around the original steady state (of ‘‘full’’ employment), depending on thedegree of steady-state inflation allowed for. There is thus an important rolefor inflation in the steady state (due to

�� n

�), since it avoids the imple-

mentation of an institutional constraint in the wage—price module at thissteady state which would, if operative at the steady state, alter the behaviorof the model in the radical fashion shown in figure 7.4.Let us now, however, come to the inclusion of endogenous growth into

the situations of fluctuations and exogenous trend growth just considered.To do this step by step, we first consider the case of a positive adjustmentparameter �� 0.2 in an otherwise unchanged environment (and againwith steady-state inflation:

��

�� 0.06 now). The trend growth rate

used by investors in their investment function thus now follows the actualrate of growth of the capital stock with a time delay in the way we haveintroduced it in section 7.2. As figures 7.5 and 7.6 suggest, these modifica-tions of the dynamics alter its dynamical features significantly, in particularwith respect to the amplitudes of the cycle.

361The road ahead

Figure 7.3 Bifurcation diagram of the 6D case for �� [0.5, 20] (see table 7.1)

Figure 7.5 shows this extended seven-dimensional dynamics over a timehorizon of 220 years, and shows a transient part of it after the steady-statevalues have been shocked at time t� 1 by a 10 percent increase in themoney supply. The most important thing in these figures is the movement(shown bottom right) of the ‘‘animal spirits’’ rate , and the correspondingmovement in the employment rate V.Figure 7.6 extends the time horizon shown to 1,000 years, and shows that

the irregularity observed in the first 220 years is not removed from thedynamics through this large time horizon. We stress once again thatassuming high adjustment speeds for all prices and quantities (as well as thekink in the Phillips curve) is responsible for the amplitudes we observe inthe cycles shown (see figure 7.6). From an empirical point of view, one maytherefore be inclined to reduce the corresponding parameters in size.However, the fact that in theory there are often situations where theseadjustment parameters are set equal even to infinitymakes it worthwhile toconsider what in fact comes about in situations of flexible price andquantity adjustment.


Figure 7.4 Downwardly rigid money wages at the inflationless steady state (table7.1 with

��

�� 0.05 (� n))

The next step now is to allow also the rate of natural growth to adjust tothe medium-run growth dynamics of the model by making it respond witha delay to the rate that prevails at each moment in time. The data relevantto figures 7.7 and 7.8 are summarized in table 7.2.Figure 7.7 shows (bottom right) that the natural rate of growth n is now

also moving in time, following the movement of the rate in an adaptivefashion. Observe also that the NAIRU-based rate V� is still assumed asconstant in this simulation of endogenous growth. Note finally that theamplitude of the fluctuations shown are somewhat larger than before.Figure 7.8 considers the behavior of this dynamical system after the

removal of its transient part after 1,000 years and for a period of 300 years.It shows that the presently considered situation in fact eventually gives risesolely to a somewhat complicated limit cycle behavior (with an indicationof period-doubling processes). Note also that there is now hysteresis visiblein the evolution of the rate n which stays below its initial steady state valueof 0.05 in the time series shown.

363The road ahead

Figure 7.5 Phase plots and times series representations of endogenous ‘‘animalspirits’’ over a time horizon of 220 years (7D case) (see table 7.1)

The above type of limit-cycle behavior is confirmed by the bifurcationdiagram in figure 7.9, drawnwith respect to the parameter�

�� [0.5, 5]. This

diagram again suggests that the attracting set is (for each ��) little more

than a somewhat complicated limit cycle (at least for most values of theparameter �

�). Therefore, a similar sort of ‘‘complex’’ dynamics over the

first 500 years may possess very different attracting sets when the transientbehavior of the dynamics is removed.Increasing the speeds of adjustment �

��and �

��of wages from 2.5 to 6,

however, leads us back to complex dynamics (via period doublings) in theconsidered situation (�

�� 0.1 in addition) as the plots in figure 7.10 show.

These are based on a transient period of 300 years (not shown), and asimulation run thereafter for 2,000 years. Fluctuations in employment, the


Figure 7.6 Phase plots and times series representations of endogenous ‘‘animalspirits’’ over a time horizon of 1,000 years (7D case)

wage share, and rates of growth are now considerably larger than before,but still within an economically viable domain. They again exhibit theirregular and now extremely asymmetric pattern in employment (and nowalso growth) we have already observed in figures 7.1 and 7.2, and they alsoagain show that there is hysteresis present in evolution of the rates ofgrowth and n. Note finally that the rate of price inflation can be minus 20percent during the course of the large fluctuations shown, and that thetrend growth rate of the capital stock is sometimes slightly negative.Let us now turn to an endogenous determination of the NAIRU-based

rate V� , and consider this situation first for growth rates and n that aregiven exogenously. We thus assume now as (modified) parameter values�v� � 0.1 and ��

�� 0 and as adjustment speeds of wages �

��and �

��equal to 1.5 (the relevant parameter set is displayed in table 7.3). In this case

365The road ahead

Figure 7.7 Phase plots and times series representations of endogenous ‘‘naturalgrowth’’ over a time horizon of 220 years (8D case) (see table 7.2)

Table 7.2.

s�� 0.8, �� 0.1, y�� 1, x� 2, n� � 0.05, n

��

�� 0.05.

h�� 0.1, h

�� 0.05, i

�� 0.25, i

�� 0.5.

��

� 2.5, ��

� 2.5, �� 1,

�� 0.5,

�� 0.5.

�� 0.6, �� 1, n�� 0, n�� 1, V� 1.

�� 0.1, �� 0.2, �v� � 0, �

�� 2, �

�� 2.

�� 0.06,

�� 0.06, t�� 0.08, �

��

�� 0.


Figure 7.8 Phase plots and times series representations of endogenous ‘‘naturalgrowth’’ after a transient period of 1,000 years (8D case) (see table 7.2)

we find, despite an inflationary steady-state situation, that the economybecomes more and more depressed. This is so since low employment ratesdeskill unemployed workers, thereby reducing the ‘‘natural’’ rate V� , whichin turn permits lower and lower actual rates of employmentV with respectto the working of the labor market. As figure 7.11 suggests, there is no realend to this process above the level of zero employment. In this case we thusget the result that each succeeding depression is more severe than thepreceding one, with no sign that this process will ever come to a standstill.Decreasing the adjustment speed of wages further, however, can alter

this situation and remove the downturn trend in the model at some laterpoint in time. This is shown in figure 7.12, where the adjustment speeds �

��of money wages have both been reduced to the value 1. Here, the model

367The road ahead

Figure 7.9 Bifurcation diagram for �� [0.5, 5] in the case of endogenous growth

(8D case)

converges to a very low level of the actual as well as the NAIRU-based rateof employment, V,V� � 0.63 approximately, with persistent cyclical move-ments on the way down to this level and damped fluctuations thereafter. Asother simulations have shown, there is considerable dependence on initialconditions (the size and direction of the monetary shock) in the presentsituation.Finally, we consider the situation where all three rates V� , , and n are

determined endogenously (figure 7.13��). Here, too, we get a long period ofcyclical downturns now with respect to employment and growth which,however, as in the previous figure, comes to a halt after approximately 350years, again with fluctuations that die out when the floor in the movementof the rate V� has been reached. We stress here that the long-run features ofthe dynamics are not close to empirically observed figures. Nevertheless,the figures drastically exemplify how downturns that are longer thanupturns, due to the asymmetry in the money-wage Phillips curve that is

�� The parameters that are set to new values in this simulation run are ��

� ��

� 2, �� 5,

�� v� � 0.1, �� 0.05,

�� 0.66.


Figure 7.10 High adjustment speeds of wages and the occurrence of ‘‘complex’’dynamics (8D case)

Table 7.3.

s�� 0.8, �� 0.1, y�� 1, x� 2, n� � 0.05, n

��

�� 0.05.

h�� 0.1, h

�� 0.05, i

�� 0.25, i

�� 0.5.

��

� 1.5, ��

� 1.5, �� 1,

�� 0.5,

�� 0.5.

�� 0.6, �� 1, �� 0, �v� � 0.1, �

�� 2, �

�� 2.

n�� 0, n�� 0, V� 1,

�� 0.1,

�� 0.1, t�� 0.08, �

��

�� 0.

369The road ahead

Figure 7.11 Phase plots and times series representations of an endogenousdetermination of the NAIRU-based rate V� (7D case) (see table 7.3)

operative here in the downturns, can drag the rates of employment and therates of growth down to levels that must be considered as highly problem-atic if not catastrophic. Such situations of self-enforcing depressions mayhave characterized to some extent the period of growth slowdown thatfollowed the sixties and early seventies.Let us contrast this result with a situation where the economy fluctuates

in such a mild way around its inflationary steady state that the kink in themoney-wage Phillips curve does not become operative. As figure 7.14 thenshows, there are then no longer self-enforcing downturns, but there is herein fact a slight increase in the steady-state rate of employment, due to thepath dependency of the natural rate of growth and employment. Such asituation may be (loosely) compared with the development in the sixtiesand seventies where in fact the growth rate of money wages did not


Figure 7.12 Phase plots and times series representations of an endogenousdetermination of the NAIRU-based rate of employmentV� (7D case) (table 7.3 with��

� ��

� 1)

approach the value zero, where the kink in the Phillips curve would havebecome operative.Finally, it is of interest to see what would have happened in the situation

considered in figure 7.13 if the money-wage Phillips curve were not kinked,so that money wages would as easily fall in situations of underemploymentas they rise in the opposite case. As figure 7.15 shows, the economy thenbecomes very explosive and would already be nonviable after a period ofabout twelve years. Again, the floor in the development of money wages isof decisive importance for the viability of the economy (though it iscertainly not the only means by which such viability is achieved). The costof this viability-generating mechanism is, however, obvious from what hasbeen said in this section. Nevertheless, advocating downward flexibility for

371The road ahead

Figure 7.13 Phase plots and times series representations of an endogenousdetermination of ‘‘natural’’ rates of employment and of growth (9D case)

the behavior of money wages to be the better alternative and sound advicefor economic income policies appears to be the opposite of the truth in thedynamicmodel considered in this chapter. This example of a dramatic lackof economic viability ends our numerical investigation of the KT modelwith endogenous long-run employment and growth.

7.7 Summary and directions for future research

In this book, we have provided a systematic theory of endogenous businessfluctuations and growth with a hierarchical structure of integrated macro-dynamical models where each subsequent model type removed one or


Figure 7.14 Phase plots and times series representations of small fluctuations inthe level of economic activity (9D case) (�

��

�� 0.3)

more important limitations of the model preceding it. We have also shownthat our analysis of the long run of monetary growth requires neither agiven ‘‘natural’’ rate of growth, nor a given income distribution, nor aclosure from the side of the market for goods via independent savings andinvestment behavior. The discussion of such closures of theories of capitalaccumulation provided in particular byMarglin (1984b) andDutt (1990) istherefore no longer applicable to the general type of monetary growththeory of this book. Nevertheless, the growth models developed by Duttand Marglin, see Dutt (1984, 1990, 1992), Marglin (1984a,b) and Marglinand Bhaduri (1991), are an important reference point for the workingmodel of this book, and are investigated from this perspective in Flaschel(1998d).Dore (1993), in his book on the macrodynamics of business cycles,

373The road ahead

Figure 7.15 Phase plots and times series representations of an endogenousdetermination of ‘‘natural’’ rates of employment and of growth (9D case) withoutextrinsic nonlinearities (�

��

�� 1.5, �

�� 10, �

��

�� 0.1, �� 0.2,

��

�� 0.05)

distinguishes basically three different approaches: The new Classical one,the new Keynesian one, and the endogenous cycle approach, where it ismaintained that business cycles are inherent to a free-enterprise economy.In this respect, our approach is definitely of the third type, though themodules we have employed on the various stages toward the derivation ofour working Keynesian model of monetary growth can surely be refor-mulated, modified, or refined from the new Keynesian perspective andothers. Our approach furthermore exhibits the components discussed byDore (1993, part III) as essential aspects for the endogenous business-cycleapproach: The Kaldor (1940) trade cycle mechanism, the Benassy (1986a)IS—LM approach to labor market dynamics and the Goodwinian (1967)growth cycle model.


In this respect we have, on the one hand, refined Kaldor’s dynamicmultiplier process toward a Metzlerian (1941) treatment of goods-marketdisequilibriumand output (plus inventory) adjustments, Benassy’s sluggishwage adjustment toward a sluggish adjustment of bothwages and prices, asin Rose (1990), and Goodwin’s growth dynamics toward a monetarygrowth dynamics that generalizes the Keynes—Wicksell approach of Rose(1967). On the other hand, we have only occasionally made use of theextrinsic nonlinearities employed by Kaldor (1940), Benassy (1986a), orRose (1967) (and others), since we wanted to concentrate on the intrinsicnonlinearities at first and their implications for the generation of en-dogenous damped, explosive, or persistent business fluctuations. At theend of the book we have, however, stressed a crucial and very basicextrinsic nonlinearity (not unrelated to the labor-market assumptionsmade by Benassy, Goodwin, and Rose), namely, the kinked Phillips curvewhich removes the possibility of wage deflation from the business cycle inthe simplest way possible.We saw there that the exclusion of nominal wagedeflation had dramatic consequences for the viability of the consideredmonetary growth dynamics.��In line with Benassy’s (1986a) treatment of the dominance of the Key-

nesian regime in the theory of business fluctuations, we have used theKeynesian effective demand regime as determining the short-run positionof the economy in each moment of time, and this for the basic Keynesianprototype model of chapter 4 and all models following it. There are avariety of mechanisms in a free-enterprise economy that, taken together,guarantee this outcome.�� The discussion of regimes other than the Key-nesian one — see Benassy (1996b), Dore (1993, ch. 10), and also Snowdon,Vane, and Wynarczyk (1994, 3.5) — thus does not apply to the Keynesianmonetary growth theory here developed. In view of chapter 3 in Snowdon,Vane, andWynarczyk (1994) we would, however, accept that our approachto monetary growth dynamics belongs to the traditional Keynesian one�of such growth dynamics, as, for example, that of Turnovsky (1977a), butnow on the basis of a full scenario of price, wage, output, and employmentadjustment processes and more.As is obvious fromDore (1993), Snowdon, Vane, andWynarczyk (1994),

Orphanides and Solow (1990), and many other surveys on macrodynamictheory, the approach we are offering in this book has been very muchneglected in the literature. There are few formalized contributions to

�� Nonlinearities in investment behavior not unrelated to those in Kaldor (1940) have beendiscussed in chapters 3 and 4 in their role of generating persistent real or even monetarygrowth cycles (see Skott 1991, however, on the difficulties of incorporating Kaldor’s tradecycle theory into the context of a growing economy).

�� These have been or could be incorporated into our monetary growth framework.� These have rarely been considered in the literature.

375The road ahead

a Keynesian theory of monetary growth in general, whether of neo-Keynesian, new Keynesian or post-Keynesian type.� We hope in thisbook to have laid the foundations of such a theory, on the basis of whichthe contributions of these various schools of thought can be evaluated and,when considered as important, be integrated into the present generalframework of the dynamics of Keynesianmonetary growth, based on workby Kaldor (1940, 1956), Metzler (1941), Goodwin (1967), Rose (1967, 1990),Tobin (1975), Turnovsky (1977a), Malinvaud (1980), Sargent (1987, part I),and their discussion in the literature. To integrate these partial views oneconomic dynamics into a consistent whole was one of the main aims ofthis book, and will continue to be our aim with respect to further exten-sions of the framework presented here.��Let us therefore review in more detail what has been done in this book

and what is required of future work in this framework.We have consideredin this final chapter a more refined treatment of the labor market, outsideand inside the firm, and have endogenized trend growth in the context ofthe KT model of the preceding chapter, i.e., in the presence of a simple,sluggish adjustment process of quantities besides the sluggish adjustmentof wages and prices. We have seen that quite new dynamic patterns areestablished through these extensions of the KTmodel of monetary growth.In this way, the final step of our hierarchy of Keynesian models of monet-ary growth was reached, though so far not in such a way that the variousextensions, presented in chapters 4—6 and this chapter, have all beencombined into one single large model. These extensions have concernedfactor substitution and technological change, wage taxation and forward-looking expectations of p-star type, averages of wage and price inflation,Metzlerian inventory adjustment processes, and, in this chapter, en-dogenous long-run rates of growth and employment and further labor-market adjustment processes. The obtained structural form of our Key-nesian model of monetary growth, when everything is put together, istherefore already of a fairly advanced type, since it allows for

• various types of delayed adjustment processes in prices, wages, quanti-ties, inventories, and employment;

• delayed adjustments of inflationary expectations of households andfirms (with forward- and backward-looking components) and in salesexpectations of firms;

• under- or overutilization of the workforce as well as the capital stock,with demand pressure effects on wage inflation and price inflation,respectively;

� See again Dore (1993), Snowdon, Vane, andWynarczyk (1994) for details on these schoolsof thought. �� See Chiarella et al. (1998, 1999), Chiarella and Flaschel (1998f ).


• smooth factor substitution, again coupled with under- or overemploy-ment of the capital stock, where one interprets the marginal wage costbarrier of output expansions as a principle that (when approached andpassed) speeds up investment and prices, but does not limit productionin the strict sense of non-Walrasian temporary equilibrium theory;

• aggregate investment depending on both relative profitability and therate of capacity utilization;

• Harrod-neutral technical change coupled with an endogenous determi-nation of the trend term of investment and of labor force growth;

• taxation of labor and capital income categories (but not yet taxation offirms or of wealth, or value added taxes, payroll taxes, and the like);��

• endogenous determination of the long-run rate of employment (but notyet of the long-run rate of capacity utilization).

Of course, all these aspects or extensions of the basic Keynesian prototypemodel of monetary growth have to be integrated with each other into aconsistent whole (to be done in a fairly obvious way as chapters 4—7 shouldhave made clear) when one wants to provide a truly general (integrated)model of Keynesian dynamics of a growing monetary economy. Further-more, such a general theoretical framework and model should now beready for macroeconometric model building and its application to actualeconomies which, in addition, demands that it must be reformulated as anopen economy. Providing such a model type for open economies surelyindicates one strand for future progress in the modeling of Keynesianmonetary growth with both underutilized labor and capital, and its appli-cation to the study of actual economies. Here in fact work in progressalready exists, see Chiarella et al. (1998, 1999), which when finished willprovide exactly this type of modeling for small open, large open, andinteracting open economies, including detailed comparisons with macro-econometric models of this type as the one of Powell and Murphy (1997)for the Australian economy.Yet, there are further aspects of our general Keynesian theory of monet-

ary growth that may require a significant overhaul of the modules we haveemployed so far for describing this approach to growth theory, in particu-lar in view of the numerous contributions to macrodynamic theory of therecent past.� There is for example the approach to the theory of unemployment by

Layard, Nickell, and Jackman (1991) and also by Carlin and Soskice (1990)(to some extent related to earlier, but nevertheless significantly different,

�� See Chiarella et al. (1998) for the extensive treatment of government taxation and transferswith respect to workers, asset holders, and firms.

� A to some extent incomplete survey on this literature is provided by Snowdon, Vane, andWynarczyk (1994).

377The road ahead

work by Rowthorn 1980) with, on the one hand, their thorough microfoundations of wage- and price-setting behavior, but, on the other hand,their restricted modeling of the macroeconomic feedback structure (whichallows for the operation of the stabilizing Keynes effect solely). Neverthe-less, these authors have surely provided an important alternative of thedynamic interaction of wages and prices as we have formulated it so far.��We add that Rowthorn’s (1980, ch. 6) original model of the distributionalconflict and inflation in a monetary framework is reformulated and con-trasted with the fundamental monetarist model of inflation and unemploy-ment in Flaschel (1993, ch. 5) and Flaschel and Groh (1996a, ch. 4).There is, furthermore, extensive work by Skott (1989a,b, 1991) which

treats effective demand and the conflict over income distribution from aperspective not unrelated to the Keynes—Wicksell monetary growth modelof this book, yet integrates this perspective with post-Keynesian andneo-Marxian approaches of the literature.��We thus see that the Keynes—Wicksell approach to a supply-side determined Keynesian theory of effec-tive demand and growth and its implications for the theory of inflation isfar from being a subject of the past, but in fact is further refined through theintroduction of a micro-founded output expansion function into the his-torically first formulation of Keynesian monetary growth dynamics.There is, as another example, the work by Lance Taylor (see for example

Taylor 1983, 1984, 1991), with the stress it lays on financial markets as wellas on the distributional conflict and its impacts on output and inflation (seealso Dutt 1984, 1990, 1992 for a treatment of this latter topic in the contextof growing economies). A fairly general formulation of the role of financialmarkets along Tobinian lines is furthermore provided in Franke andSemmler (1999), extending earlier approaches of Franke and Semmler onfinancial markets, an approach that in our view can be integrated with theanalysis presented in this book in a fairly direct way. Aspects of thestructural approach to macrodynamics of Taylor, Dutt, and others, whichhas also been applied to less developed countries, are brought together inthe volume edited by Epstein and Gintis (1995), where various theories ofsustainable economic growth are discussed and evaluated, also from theempirical perspective.There is the literature on endogenous growth which has revitalized the

various approaches on the production of human knowledge and qualityimprovements, see Barro and Sala-i-Martin (1995) and Aghion andHowitt(1998), for example. This type of endogenous growth theory is reconsidered

�� See also Dixon and Rankin (1995).�� See also Flaschel, Franke, and Semmler (1997) on this approach and its relationships to

models of AD—AS growth and Flaschel (1998c) for a treatment of his work from theperspective of the general Keynes—Wicksell model we have developed and investigated inchapter 3 of this book.


from a Keynesian perspective in Flaschel and Groh (1996a).There is, furthermore, the vast literature on real business cycle models,

for closed as well as open economies, on their theoretical relevance andtheir empirical validity, and their extension to include nominal or realrigidities and monopolistic competition (see Henin 1995 for example).This varied set of approaches, as well as further ones, to modern macro-

dynamic model building are explored, extended, to some extent integrated,and put into perspective in Chiarella et al. (1999). This work in particularshows how various supply bottlenecks can be integrated into the workingKeynesian model of the present book, thereby adding a complete andcoherent treatment of large business swings which can run into such supplybottlenecks. The Keynesian dynamical system (still of traditional type) soobtained thus then allows for regime switching in particular circumstances.Its strengths and weaknesses are comparedwith the competing approachesof the new Classical and the new Keynesian variety. Furthermore, recentcontributions of Keynes—Wicksell type are also treated extensively, gener-alized, and compared with the Keynes—Metzler approach to monetarygrowth, as well as post-Keynesian ones and their particular treatments ofthe conflict about income distribution, financial markets, and more. Flas-chel (1998f), therefore, puts into perspective what has been achieved in thepresent volume by its systematic reconstruction and hierarchical develop-ment of integrated Keynesian models of monetary growth.There is, finally, the literature on macroeconometric model building (see

Bodkin, Klein, and Marwah 1991 and Whitley 1994 for recent surveys onthis literature). We have already stressed in this regard the Murphy modelfor the Australian economy as it is presented in Powell andMurphy (1997).This model of a small open economy is fairly close in its theoreticalstructure to the (closed economy) working model, and its various exten-sions, that we have introduced and analyzed in this book and which will beanalyzed in our future work in the way we have indicated above (seeChiarella et al. 1998). There are, of course, many other important macro-econometric models of (larger) open economies, such as the Fair-model fortheUS economy (see Fair 1994 for a detailed presentation and review of hiswork on macroeconometric model building [also in the multi-countrycontext], and Godley and Anyadike-Danes 1987, Godley 1998; see alsoGodley and Cripps 1983 for a textbook presentation of such an approach,and the Westphal model for the German economy in Dieckmann andWestphal 1995). Thesemodels are, however, generallymuch larger than theMurphy model for the Australian economy, as presented in a very detailedway in Powell and Murphy (1997), and therefore also less close to thetheoretical working model developed in this book and to be developed inour future work.In view of the foregoing discussion, further extensions of the working

379The road ahead

Keynesian monetary growth model of this book thus could and willinvolve the following aspects whichwould lead to a further improvement ofits various sectors and the modules:

1 The household sector, and there in particular:(a) The distinction between short-term and long-term bonds by which

cash-management processes and the treatment of long-term creditrelationships can be distinguished from each other, implying the exist-ence of a term structure of interest rates for the model.

(b) A more advanced and less ‘‘tranquil’’ asset structure on the assetmarkets and an explicit treatment of expected capital gains for equitiesas well as long-term bonds.

(c) The existence of chartist and fundamentalist groups of asset owners,and thus of heterogeneous expectation formation processes.

(d) Imperfect substitutability between all assets and the inclusion of adynamic portfolio balance approach as in Turnovsky (1995).

(e) A more advanced theory of money demand and of processes of cashmanagement.

(f ) More advanced concepts of perceived disposable income (as these were,for example, considered in chapter 2).

(g) Positive savings out of wages (s�� 0) and further differentiated saving

habits and their implications for the accumulation of financial assets.(h) Intertemporal aspects and micro foundations of household behavior.

2 The sector of firms, and there in particular:(a) More complex technological relationships in production.(b) A more elaborate theory of investment and of its financing by retained

earnings or the issuing of new equities or by bank loans.(c) Debt financing and money holdings of firms.(d) Further intertemporal aspects and micro foundations of the behavior

of firms.(e) An endogenous treatment of the ‘‘normal’’ rate of capacity utilization

introduced in chapter 4.(f ) Micro foundations and modifications of the price-level Phillips curve

with its demand-pull and cost-push components.(g) Micro foundations of quantity adjustments, and in particular the in-

ventory adjustment behavior of firms.

3 The government sector, and there in particular:(a) The introduction of more elaborate fiscal and monetary (and interest

rate) policy rules.


(b) An active debt policy of the government (and marketing difficulties inthe sale of new bonds B� ).

(c) Further intertemporal aspects in the behavior of the government andthe central bank.

(d) Problems for policy coordination.

4 The wage—price module of the models:(a) Further improvements in the price and wage adjustment rules to be

based on micro-founded target levels for prices as well as nominalwages.

(b) More advanced expectations formation mechanism and forecastingrules of backward- and/or forward-looking type (chartist time seriesmethods and fundamentalist theory based types of behavior).

5 Discrete time analysis:(a) Discrete time formulations of the considered models with an exact

dating of all activities and the expectations to which they give rise.(b) A more thorough discussion and formulation of the various budget

restraints in discrete time as well as in the continuous time limit.(c) The role of lags in economic dynamics.(d) The integration of stochastic elements into certain behavioral equa-

tions,and thus the inclusion of stochastic shocks into the endogenousand so far deterministic theories of monetary growth, inflation, andbusiness fluctuations.

6 Open economies:(a) The treatment of open economies (small, large, and interacting ones)

with trade in goods as well as in financial assets which integrates theachievements in points 1 to 5 above.

(b) A detailed comparison with, and theoretical reflection of, prominenteconometric models of such economies.

(c) Monetary and fiscal policy issues and feedback policy rules in openeconomies.

(d) The comparison of theoretical feedback analysis and the implied dy-namic patterns with those generated from macroeconometric simula-tion studies.

We have already referred to work in progress, which extends the presentlyachieved level of generality of the dynamics of Keynesian monetarygrowth, inflation and endogenous fluctuations to isolated or interactingopen economies, in Chiarella and Flaschel (1998c—f ), with respect tomacroeconometric model building in Chiarella et al. (1998) and Flaschel,

381The road ahead

Gong, and Semmler (1998), and with respect to alternative approaches toKeynesian and related macrodynamics in Flaschel (1998a—d), Chiarella etal. (1999), Flaschel (1998a—d), approaches which are there investigatedfrom the integrated perspective we have developed in this book.��These extensions include consideration of less tranquil financial markets

(which distinguish short- from long-term bonds, equities, and bank loansto firms), further supply side problems, a more advanced treatment ofheterogeneous households and their income distribution, modern govern-ment policy feedback rules,�� and a more detailed description of theemployment, pricing and investment decisions of firms. In this way weultimately hope to provide a Keynesian theory of business fluctuations,inflation, and growth which, on the one hand, is consistent in its use ofeconomic dimensions, budget restraints, and behavioral relationships andwhich, on the other hand, is general and transparent enough to provide asound basis for the understanding of structural macroeconometric modelbuilding and applied economic theory. At the same time we hope toprovide a specific answer to the views that are expressed in Blanchard(1997), Blinder (1997), Eichenbaum (1997), Solow (1997), and Taylor (1997)concerning the question ‘‘Is there a core to practical macroeconomics thatwe should all believe?’’

�� The numerical analysis of all these model types is in particular grounded on Chiarella,Flaschel, and Khomin (1998).

�� See here also Flaschel and Groh (1996b).


References

Aghion,P. and P. Howitt (1998)EndogenousGrowth Theory.Cambridge,MA:MITPress

Akerlof, G.A. and J.E. Stiglitz (1969) Capital, wages and structural unemployment.Economic Journal, 79, 269–281

Allen, C. and S. Hall (1997)Macroeconomic Modelling in a Changing World. NewYork John Wiley & Sons

Allen, H. and M.P. Taylor (1990) Chart analysis and the foreign exchange market.Review of Futures Markets, 9, 288—319

Andronov, A.A., A.A. Vitt and S.E. Chaikin (1966) Theory of Oscillators. OxfordPergamon Press

Arrowsmith, D.K. and C.M. Place (1990),Ordinary Differential Equations. LondonChapman and Hall

Asada, T. (1991) On a mixed competitive-monopolistic macrodynamic model in amonetary economy. Journal of Economics, 54, 33—53

Barro, R. (1974) Are government bonds net wealth? Journal of Political Economy,82, 1095—1117

(1994a) Macroeconomics. New York John Wiley(1994b) The aggregate-supply aggregate-demand model. Eastern EconomicJournal, 20, 1—6

Barro, R. and X. Sala-i-Martin (1995) Economic Growth.New York: McGraw-HillBenassy, J.-P. (1986a) A non-Walrasian model of the business cycle. In R. Day and

G. Eliasson (eds.), The Dynamics of Market Economies. Amsterdam: North-Holland.

(1986b)Macroeconomics. An Introduction to the Non-Walrasian Approach.NewYork Academic Press

Benhabib, J. and T. Miyao (1981) Some new results on the dynamics of thegeneralized Tobin model. International Economic Review, 22, 589—596

Blanchard, O.J. (1981) Output, the stock market, and interest rates. AmericanEconomic Review, 71, 132—143

(1997) Is there a core of usable macroeconomics? American Economic Review,Papers and Proceedings, 87, 244—246

Blanchard, O.J. and S. Fischer (1989) Lectures on Macroeconomics. Cambridge,MA: MIT Press

Blanchard, O.J. and C.M. Kahn (1980) The solution of linear difference models

383

under rational expectations. Econometrica, 48, 1305—1311Blanchard, O. and L.F. Katz (1997) What we know and do not know about the

natural rate of unemployment. Journal of Economic Perspectives, 11, 51—72Blanchflower,D.G. andA.J.Oswald (1990)Thewage curve.Scandinavian Journal of

Economics, 92, 215–235(1994) The Wage Curve. Cambridge, MA: MIT Press(1995) An introduction to the wage curve. Journal of Economic Perspectives, 9,153—167

Blinder, A.S. (1990) Inventory Theory and Consumer Behavior. Hemel Hempstead:Harvester Wheatsheaf

(1997) Is there a core of practical macroeconomics that we should all believe?American Economic Review, Papers and Proceedings, 87, 240—243

Bodkin, R., L. Klein and K. Marwah (1991) A History of MacroeconometricModel-Building. Aldershot: Edward Elgar

Brems, H. (1980) Inflation, Interest, and Growth. Lexington, MA D.C. HeathBrock, W.A. and C.H. Hommes (1997) A rational route to randomness. Economet-

rica, 65, 1059—1095Brock,W.A. andA.G.Malliaris (1989)Differential Equations, Stability andChaos in

Dynamic Economics. Amsterdam North HollandBuiter, W. (1984) Saddlepoint problems in continuous time rational expectations

models: a general method and some macrodynamic examples. Econometrica,52, 665—680

Burmeister, E. (1980)On some conceptual issues in rational expectationsmodelling.Journal of Money, Credit and Banking, 12, 217—228

Burmeister, E. and R. Dobell (1970) Mathematical Theories of Economic Growth.New York: Macmillan

Cagan, P. (1967) The monetary dynamics of hyperinflation. In Milton Friedman(ed.), Studies in the Quantity Theory of Money.Chicago: University of ChicagoPress

(1979) Persistent Inflation. Historical and Policy Essays. New York: ColumbiaUniversity Press

(1991) Expectations in the German hyperinflation reconsidered. Journal ofInternational Money and Finance, 10, 552—560

Carlin, W. and D. Soskice (1990)Macroeconomics and the Wage Bargain. Oxford:Oxford University Press

Chiarella, C. (1986) Perfect foresight models and the dynamic instability problemfrom a higher viewpoint. Economic Modelling, 3, 283—292

(1990) The Elements of a Nonlinear Theory of Economic Dynamics. Berlin:Springer Verlag

Chiarella, C. and P. Flaschel (1995) Keynesian monetary growth dynamics: themissing prototype. In J. Flemmig (ed.), Moderne Makrookonomik. Einekritische Bestandsaufnahme. Marburg: Metropolis-Verlag, 345—411

(1996a) Real and monetary cycles in models of Keynes—Wicksell type. Journal ofEconomic Behavior and Organisation, 30, 327—351

(1996b) An integrative approach to prototype 2D-macromodels of growth, price

384 References

and inventory dynamics. Chaos, Solitons & Fractals, 7, 2105—2133(1998a) Some policy experiments in a complete Keynesian model of monetarygrowth with sluggish price and quantity adjustments. To appear in W.A.Barnett, C. Chiarella, S. Keen, R. Marks, and H. Schnabl (eds.), Commerce,Complexity and Evolution. Cambridge: Cambridge University Press

(1998b) Dynamics of ‘‘natural’’ rates of growth and employment.MacroeconomicDynamics, 2, 345—368

(1998c) An integrative approach to disequilibrium growth dynamics in openeconomies. Discussion paper, University of Bielefeld

(1998d) Keynesian monetary growth dynamics in open economies. Annals ofOperations Research, 89, 35—59

(1998e) ‘‘High order’’ disequilibrium growth dynamics: theoretical aspects andnumerical features. Journal of Economic Dynamics and Control (special issue)

(1998f)DisequilibriumGrowth Dynamics in Open Economies.Heidelberg: Spring-er Verlag

(2000) Stability properties of high-dimensional AS—AD disequilibrium growthdynamics. Working paper, School of Finance and Economics, University ofTechnology, Sydney

Chiarella, C., P. Flaschel, G. Groh, C. Koper and W. Semmler (1998) ModernMacroeconometric Model Building: Theory, Numerical Analysis and Applica-tions. Bielefeld/Sydney: Book Manuscript

Chiarella, C., P. Flaschel, G. Groh and W. Semmler (1999) Disequilibrium, Growthand Labor Market Dynamics: Macroperspectives.Heidelberg: Springer Verlag

Chiarella,C., P. Flaschel andA.Khomin (1998)Numerical Simulation ofHighOrderMacrodynamic Systems: Tools and Results.University of Technology, Sydney:Book Manuscript

Chiarella, C. and Khomin (1999) Adaptively evolving expectations in models ofmonetary dynamics: the fundamentalists forward looking. Annals of Oper-ations Research, 89, 21—34

Chiarella, C. and H.-W. Lorenz (1996) The nonlinear generalized Tobin model.University of Technology, Sydney, mimeo; revised as The dynamics of thenonlinear generalised Tobin model. Working paper, School of Finance andEconomics, University of Technology, Sydney, 2000

Cross, R.B. (1987) Hysteresis and instability in the natural rate of unemployment.Scandinavian Journal of Economics, 89, 71—89

(1995) (ed.) The Natural Rate of Unemployment.Cambridge: Cambridge Univer-sity Press.

Dieckmann, O. and U. Westphal (1995) Sysifo: Ein okonmetrisches Modell derdeutschen Volkswirtschaft. Hamburg Dr. Siegel & Partner

Dixon, H.D. and N. Rankin (1995) (eds.) The New Macroeconomics: ImperfectMarkets and Policy Effectiveness. Cambridge: Cambridge University Press

Dore,M. (1993)TheMacrodynamics of Business Cycles: A Comparative Evaluation.Oxford: Blackwell

Dornbusch,R. (1976)Expectations and exchange rate dynamics. Journal of PoliticalEconomy, 84, 1161—1176

385References

Dutt, A. (1984) Stagnation, income distribution, and monopoly power. CambridgeJournal of Economics, 8, 25—40

(1990) Growth, Distribution, and Uneven Development. Cambridge: CambridgeUniversity Press

(1992) On the long-run stability of capitalist economies: implications of a modelof growth and distribution. University of Notre Dame, mimeo

Eichenbaum, M. (1997) Some thoughts on practical stabilization policy. AmericanEconomic Review, Papers and Proceedings, 87, 236—239

Epstein, G. and H. Gintis (1995)Macroeconomic Policy after the Conservative Era:Studies in Investment, Saving and Finance. Cambridge: Cambridge UniversityPress

Fair, R. (1994) Testing Macroeconometric Models. Cambridge, MA: HarvardUniversity Press

(1997a) Testing theNAIRUmodel for the United States. Yale University, mimeo(1997b) Testing the NAIRU model for 27 countries. Yale University, mimeo

Ferri, P. and E. Greenberg (1989) The Labor Market and Business Cycle Theories.Heidelberg: Springer Verlag

Fischer, S. (1972) Keynes—Wicksell and neoclassical models of money and growth.American Economic Review, 62, 880—890

Flaschel, P. (1984) Some stability properties of Goodwin’s growth cycle model.Zeitschrift fur Nationalokonomie, 44, 63—69

(1993) Macrodynamics: Income Distribution, Effective Demand and CyclicalGrowth. Bern: Verlag Peter Lang

(1998a) On the dominance of the Keynesian regime in disequilibrium growththeory: a note. To appear in Journal of Economics

(1998b) Disequilibrium growth theory with insider/outsider effects: a note. Toappear in Structural Change and Economic Dynamics

(1998c) Corridor stability and viability in economic growth. Discussion paper,University of Bielefeld

(1998d) Keynes—Marx and Keynes—Wicksell models of monetary growth: aframework for future analysis. To appear in Review of Political Economy

Flaschel, P., Franke, R. andW. Semmler (1997)DynamicMacroeconomics Instabil-ity, Fluctuations and Growth in Monetary Economies. Cambridge, MA: MITPress

Flaschel, P., Gong, G. and W. Semmler (1998) A Keynesian based econometricframework for studying monetary policy rules. University of Bielefeld, mimeo

Flaschel, P. and G. Groh (1995) The Classical growth cycle: reformulation,simulation and some facts. Economic Notes, 24, 293—326

(1996a) Keynesianische Makrookonomik: Unterbeschaftigung, Inflation undWachstum. Heidelberg: Springer Verlag

(1996b) The stabilizing potential of policy rules in Keynesian monetary growthdynamics. Systems Analysis – Modelling – Simulation, 23, 39—72

Flaschel, P. and R. Sethi (1996) Classical dynamics in a general model ofKeynes—Wicksell type. Structural Change and Economic Dynamics, 7, 401—428

(1999) The stability of models of monetary growth: implications of nonlinearity.

386 References

Economic Modelling, 16, 221—233Flood, D. and P. Lowe (1995) Inventories and the business cycle.Economic Record,

71, 27—39Franke, R. (1992a) Stable, unstable, and cyclical behaviour in a Keynes—Wicksell

monetary growth model. Oxford Economic Papers, 44, 242—256(1992b) Inflation and distribution in a Keynes—Wicksell model of the businesscycle. European Journal of Political Economy, 8, 599—624

(1996) A Metzlerian model of inventory growth cycles. Structural Change andEconomic Dynamics, 7, 243—262

Franke, R. and T. Asada (1993) A Keynes—Goodwin model of the business cycle.Journal of Economic Behavior and Organization, 24, 273—295

Franke, R. and T. Lux (1993) Adaptive expectations and perfect foresight in anonlinear Metzler model of the inventory cycle. Scandinavian Journal ofEconomics, 95, 355—363

Franke, R. and W. Semmler (1999) Bond rate, loan rate and Tobin’s q in atemporary equilibrium model of the financial sector. To appear in Met-roeconomica

Frankel, J.A. and K. Froot (1987) Using survey data to test standard propositionsregarding exchange rate expectations.AmericanEconomicReview, 77, 133—153

(1990) Chartists, fundamentalists and trading in the foreign exchange market.American Economic Review, 80, 181—185

Franz,W. (1990)HysteresisEffects in EconomicModels.Heidelberg:PhysicaVerlagFriedman, B. and F. Hahn (1990) (eds.) Handbook of Monetary Economics.

Amsterdam: North-HollandFujino, S. (1974) A Neokeynesian Theory of Inflation and Economic Growth.

Heidelberg: Springer VerlagGalbraith, J.K. (1997) Time to ditch the NAIRU. Journal of Economic Perspectives,

11, 93—108.Gale, D. (1983)Money in Disequilibrium. Cambridge: Cambridge University PressGandolfo, G. (1997) Economic Dynamics. Heidelberg: Springer VerlagGantmacher, F.R. (1959), Applications of the Theory of Matrices. New York:

Interscience PublishersGeorge, D.A.R. and L.T. Oxley (1985) Structural stability and model design.

Economic Modelling, 2, 307—316Godley, W. (1998) Money and credit in a Keynesian model of income determina-

tion. Jerome Levy Economics Institute, New York: mimeoGodley, W. and M. Anyadike-Danes (1987) A stock-adjustment model of income

determination with inside money and private debt with some preliminaryempirical results for the United States. In M. deCecco and J.-P. Fitoussi (eds.),Monetary Theory and Economic Institutions. New York: St. Martin’s Press

Godley,W. and F.Cripps (1983)Macroeconomics.Oxford:OxfordUniversity PressGoldman, S.M. (1972) Hyper inflation and the rate of growth in the money supply.

Journal of Economic Theory, 5, 250—257Goodwin,R.M. (1967) A growth cycle. In C.H. Feinstein (ed.),Socialism, Capitalism

and Economic Growth. Cambridge: Cambridge University Press, 54—58

387References

Gordon, R. (1997) The time-varying NAIRU and its implications for economicpolicy. Journal of Economic Perspectives, 11, 11—32

Grandville, O. de la (1986) Dynamics and stability in IS—LMmodel: a reexamina-tion. Journal of Macroeconomics, 8, 31—41

Groth, C. (1988) IS—LM dynamics and the hypothesis of combined adaptiveforward-looking expectations. In P. Flaschel and M. Kruger (eds.), RecentApproaches to Economic Dynamics. Bern: Verlag Peter Lang, 251—265

(1992) Keynesian-monetarist dynamics and the corridor: a note. University ofCopenhagen, mimeo

(1993) Some unfamiliar dynamics of a familiar macro model. Journal ofEconomics, 58, 293—305

Guckenheimer, J. and P. Holmes (1983)Nonlinear Oscillations, Dynamical Systemsand Bifurcations of Vector Fields. Heidelberg: Springer Verlag

Hadjimichalakis, M.G. (1971a) Money, expectations and dynamics: an alternativeview. International Economic Review, 12, 381—402

(1971b) Equilibrium and disequilibrium growth with money: the Tobin models.Review of Economic Studies, 38, 457—479

(1973) On the effectiveness of monetary policy as a stabilization device.Review ofEconomic Studies, 40, 561—570

(1981a) The Rose—Wicksell model: inside money, stability and stabilizationpolicies. Journal of Macroeconomics, 3, 369—390

(1981b) Expectations of the ‘‘myopic perfect foresight’’ variety in monetarydynamics. Journal of Economic Dynamics and Control, 3, 157—176

Hadjimichalakis,M.G. andK.Okuguchi (1979) The stability of a generalizedTobinmodel. Review of Economic Studies, 46, 175—178

Hahn, F. and R. Solow (1995) A Critical Essay on Modern Macroeconomic Theory.Cambridge, MA: MIT Press

Hairer E., S.P. Nørsett and G. Wanner (1987) Solving Ordinary DifferentialEquations I: Nonstiff Problems. Heidelberg: Springer Verlag

Hayakawa, H. (1979) Real purchasing power in the neoclassical growth model.Journal of Macroeconomics, 1, 19—31

(1983) Rationality of liquidity preferences and the neoclassical growth model.Journal of Macroeconomics, 5, 495—501

(1984) A dynamic generalization of the Tobin model. Journal of EconomicDynamics and Control, 7, 209—231

Heap, H.S. (1980) Choosing the wrong natural rate: accelerating inflation ordecelerating employment and growth. Economic Journal, 90, 611—620

Henin, P.-Y. (1995) (ed.)Advances in Business Cycle Research.Heidelberg: SpringerVerlag

Henin, P.-Y. and P.Michel (1982)Croissance et accumulation en desequilibre. Paris:Economica

Hicks, J. (1974) Real and monetary factors in economic fluctuations. ScottishJournal of Political Economy, 21, 205—214

Hirsch, M.W. and S. Smale (1974) Differential Equations, Dynamical Systems, andLinear Algebra. New York: Academic Press

388 References

Ito, T. (1980) Disequilibrium growth theory. Journal of Economic Theory, 23,380—409

Iwai, K. (1981) Disequilibrium Dynamics. New Haven: Yale University PressJohnson, H.G. (1966) The neoclassical one-sector growth model: a geometrical

exposition and extension to a monetary economy. Economica, 33, 265—287(1967a) Money in a neoclassical one sector growth model. In H.G. Johnson,Essays in Monetary Economics. Cambridge, MA: Harvard University Press

(1967b) The neutrality of money in growthmodels: a reply. Economica, 34, 73—74Jones, H.G. (1975) An Introduction to Modern Theories of Economic Growth.

London: Thomas Nelson & SonsKaldor, N. (1940) A model of the trade cycle. Economic Journal, 50, 78—92(1956) Alternative theories of distribution. Review of Economic Studies, 23,83—100

Karakitsos, E. (1992) Macrosystems: The Dynamics of Economic Policy. Oxford:Basil Blackwell

Kearney, C. and R. MacDonald (1985) Asset markets and the exchange rate: astructural model of the sterling—dollar rate, 1972—1982. Journal of EconomicStudies, 12, 3—20

Keynes, J.M. (1936) The General Theory of Employment, Interest and Money.NewYork: Macmillan

Khibnik, A., Y. Kuznetsov, V. Levitin and E. Nikolaev (1993) Continuationtechniques and interactive software for bifurcation analysis of ODEs anditerated maps. Physica D, 62, 360—371

Klundert, T. van de and A. van Schaik (1990) Unemployment persistence and theloss of productive capacity: a Keynesian approach. Journal of Macro-economics, 12, 363—380.

Kuh, E. (1967) A productivity theory of the wage levels — an alternative to thePhillips curve. Review of Economic Studies, 31, 333—360

Kuznetsov, Y.A. (1995) Elements of Applied Bifurcation Theory. Heidelberg:Springer Verlag

Laxton, D., D. Rose and D. Tambakis (1997) The US Phillips curve: the case forasymmetry. Mimeo

Layard, R., S. Nickell and R. Jackman (1991) Unemployment: MacroeconomicPerformance and the Labor Market. Oxford: Oxford University Press

Leslie,D. (1993)AdvancedMacroeconomics:BeyondIS/LM.London:McGraw-HillLindbeck,A. andD. Snower (1988)The Insider–OutsiderTheory of Employment and

Unemployment. Cambridge, MA: MIT PressLorenz, H.-W. (1997) Nonlinear Dynamical Economic and Chaotic Motion.Heidel-

berg: Springer VerlagLux, T. (1993) A note on the stability of endogenous cycles in Diamond’s model of

search and barter. Journal of Economics, 56, 185—196(1995) Corridor stability in the Dendrinos model of regional factor movements.Geographical Analysis, 27, 360—368

Malinvaud, E. (1980) Profitability and Unemployment. Cambridge: CambridgeUniversity Press

389References

Mankiw, G. (1990) A quick refresher course in macroeconomics. Journal ofEconomic Literature, 28, 1645—1660

Marglin, S. (1984a) Growth, distribution, and inflation: a centennial synthesis.Cambridge Journal of Economics, 8, 115—144

(1984b) Growth, Distribution, and Prices. Cambridge, MA: Harvard UniversityPress

Marglin, S. and A. Bhaduri (1991) Profit squeeze and Keynesian theory. In E. NellandW. Semmler (eds.),Nicholas Kaldor andMainstream Economics: Confron-tation or Convergence? London: Macmillan, 123—163

Marx, K. (1954) Capital, vol. I. London: Lawrence and WishartMatsumoto, A. (1995a)Non-linear structure of aMetzlerian inventory cycle model.

Niigata University, Japan, mimeo(1995b) A non-linear macro model of endogenous inventory oscillations. NiigataUniversity, Japan, mimeo

McCallum,B.T. (1983) Onnon-uniqueness in rational expectationsmodels. Journalof Monetary Economics, 11, 139—168

(1989) Monetary Economics Theory and Policy. New York: MacmillanMedio, A. (1991) Continuous-time models of chaos in economics. Journal of

Economic Behavior and Organization, 16, 133—151(1992) Chaotic Dynamics: Theory and Applications to Economics. Cambridge:Cambridge University Press

Metzler, L.A. (1941) The nature and stability of inventory cycles. Review ofEconomic Statistics, 23, 113—129

Murphy,K.M. andR. Topel (1997) Unemployment and nonemployment.AmericanEconomic Review, Papers and Proceedings, 87, 295—300

Nagatani, K. (1970) A note on Professor Tobin’s money and economic growth.Econometrica, 38, 171—175

(1978) Monetary Theory. Amsterdam: North-HollandOrphanides, A. and R. Solow (1990) Money, inflation and growth. In B. Friedman

and F. Hahn (eds.), Handbook of Monetary Economics. Amsterdam: North-Holland, 223—261

Oxley, L. and D.A.R. George (1994) Linear saddlepoint dynamics ‘‘on their head’’:the scientific content of the new orthodoxy in macrodynamics. EuropeanJournal of Political Economy, 10, 389—400

Papell, D. (1992) Exchange rate and price dynamics under adaptive and rationalexpectations: an empirical analysis. Journal of International Money andFinance, 11, 382—396

Parker, T.S. and L.O. Chua (1989) Practical Numerical Algorithms for ChaoticSystems. Berlin: Springer Verlag

Patinkin, D. (1965) Money, Interest and Prices. New York: Harper and RowPerko, M.L. (1993) Differential Equations and Dynamical Systems. Heidelberg:

Springer VerlagPhelps, E. and G. Zoega (1997) The rise and downward trend of the natural rate.

American Economic Review, Papers and Proceedings, 87, 283—289Phillips, A.W. (1958) The relation between unemployment and the rate of change of

390 References

moneywage rates in theUnitedKingdom, 1861—1957.Economica, 25, 283—299Pohjola,M. (1981) Stable, cyclic and chaotic growth: the dynamics of a discrete-time

version of Goodwin’s growth cycle model. Zeitschrift fur Nationalokonomie,41, 27—38

Powell, A. and C. Murphy (1997) Inside a Modern Macroeconometric Model: AGuide to the Murphy Model. Heidelberg: Springer Verlag

Puu, T. (1997) Nonlinear Economic Dynamics. Heidelberg: Springer VerlagRogerson, R. (1997) Theory ahead of language in the economics of unemployment.

Journal of Economic Perspectives, 11, 73—92Romer, D. (1996) Advanced Macroeconomics. London: McGraw-HillRose, H. (1966) Unemployment in a theory of growth. International Economic

Review, 7, 260—282(1967) On the non-linear theory of the employment cycle. Review of EconomicStudies, 34, 153—173

(1969) Real and monetary factors in the business cycle. Journal of Money, Creditand Banking, 1, 138—152

(1990) Macroeconomic Dynamics: A Marshallian Synthesis. Cambridge, MA:Basil Blackwell

Rowthorn, B. (1980) Capitalism, Conflict and Inflation. London: Lawrence andWishart

Saint-Paul, G. (1997) The rise and persistence of rigidities. American EconomicReview, Papers and Proceedings, 87, 290—294

Samuelson, P. and R. Solow (1960) The problem of achieving and maintaining astable price level: analytical aspects of of anti-inflation policy. AmericanEconomic Review, 50, 177—194

Sargent, T. (1987) Macroeconomic Theory. New York: Academic PressSargent, T. andN.Wallace (1973) The stability of models ofmoney and growthwith

perfect foresight. Econometrica, 41, 1043—1048Sethi, R. (1996) Endogenous regime switching in speculative markets. Structural

Change and Economic Dynamics, 7, 99—118Sidrauski,M. (1967a) Inflation and economic growth. Journal of Political Economy,

75, 796—810(1967b) Rational choice and patterns of growth in a monetary economy.American Economic Review, 57, 534—544

Sijben, J. (1977) Money and Economic Growth. Leiden: Martinus NijhoffSkott, P. (1989a) Conflict and Effective Demand in Economic Growth. Cambridge:

Cambridge University Press(1989b) Effective demand, class struggle and cyclical growth. InternationalEconomic Review, 30, 231—247

(1991) Cyclical Growth in a Kaldorian model. In E.J. Nell and W. Semmler (eds.),Nicholas Kaldor and Mainstream Economics: Confrontation or Convergence?London: Macmillan, 379—394

Snowdon, B., H. Vane and P. Wynarczyk (1994) A Modern Guide to Macro-economics: An Introduction to Competing Schools of Thought. Aldershot:Edward Elgar

391References

Solow, R. (1956) A contribution to the theory of economic growth. QuarterlyJournal of Economics, 70, 65—94

(1990) Goodwin’s growth cycle: reminiscence and rumination. In K. Velupillai(ed.), Nonlinear and Multisectoral Macrodynamics. London: Macmillan, 31—41

(1997) Is there a core of usable macroeconomics we should all believe in?American Economic Review, Papers and Proceedings, 87, 230—232

Solow, R. and J. Stiglitz (1968) Output, employment and wages in the short-run.Quarterly Journal of Economics, 82, 537—560

Staiger, D., J.H. Stock and M.W. Watson (1997) The NAIRU, unemployment andmonetary policy. Journal of Economic Perspectives, 11, 33—50

Stein, J. (1966) Money and capacity growth. Journal of Political Economy, 74,451—465

(1968) Rational choice and the patterns of economic growth in a monetaryeconomy: comment. American Economic Review, 58, 944—950

(1969) Neoclassical and Keynes—Wicksell monetary growth models. Journal ofMoney, Credit and Banking, 1, 153—171

(1970) Monetary growth theory in perspective. American Economic Review, 60,85—106

(1971) Money and Capacity Growth. New York: Columbia University Press(1982) Monetarist, Keynesian and New Classical Economics. London: BasilBlackwell

Stiglitz, J. (1997) Reflections on the natural rate hypothesis. Journal of EconomicPerspectives, 11, 3—10

Strogatz, S.H. (1994) Nonlinear Dynamics and Chaos. New York: Addison-WesleySummers, L. (1990) Understanding Unemployment. Cambridge, MA: MIT PressTaylor, J.B. (1997) A core of practicalmacroeconomics.AmericanEconomicReview,

Papers and Proceedings, 87, 233—235Taylor, L. (1983) Structuralist Macroeconomics. New York: Basic Books(1984) A stagnationist model of economic growth. Cambridge Journal ofEconomics, 9, 383—503

(1991) Income Distribution, Inflation, and Growth. Cambridge, MA: MIT PressTobin, J. (1955) A dynamic aggregative model. Journal of Political Economy, 63,

103—115(1961) Money, capital, and other stores of value. American Economic Review, 51,26—37

(1965) Money and economic growth. Econometrica, 33, 671—684(1975) Keynesian models of recession and depression. American EconomicReview, 65, 195—202

(1992) An old Keynesian counterattacks.Eastern Economic Journal, 18, 387—400(1993) Price flexibility and output stability. Journal of Economic Perspectives, 18,387—400

Tsiang, S.C. (1982) Stock or portfolio approach to monetary theory and theneo-Keynesian school of James Tobin. IHS Journal, 6, 149—171

Tu, P. (1994) Dynamical Systems. Heidelberg: Springer Verlag

392 References

Turnovsky, S.J. (1977a)Macroeconomic Analysis and Stabilization Policies. Cam-bridge: Cambridge University Press

(1977b)On the formulationof continuous timemacroeconomicmodelswith assetaccumulation. International Economic Review 18, 1—27

(1995) Methods of Macroeconomic Dynamics. Cambridge, MA: MIT PressWhitley, J. (1994) A Course in Macroeconomic Modelling and Forecasting. New

York: Harvester WheatsheafWiggins, S. (1990) Introduction to Applied Nonlinear Dynamical Systems and Chaos.

Heidelberg: Springer VerlagWolfstetter, E. (1982) Fiscal policy and the classical growth cycle. Journal of

Economics, 42, 375—393

393References

Author index

Adronov, A.A., 56Aghion, P., 378Akerlof, G.A., 143, 145, 148Allen, C., 58, 64, 338, 342Anyadike-Danes, M., 379Arrowsmith, D.K., 66Asada, T., 14, 37

Barro, R., 17, 70, 285, 378Benassy, J.P., 17, 21, 197, 198, 231, 232, 257,

258, 374, 375Benhabib, L., 13, 29, 83, 87, 99, 120, 158,

214, 250Bhaduri, A., 373Blanchard, O., 48, 64, 286, 342, 382Blanchflower, D.G., 348Blinder, A.S., 64, 294, 338Bodkin, R., 379Brems, H., 14Brock, W.A., 61, 66, 141, 148, 156Buiter, W., 47Burmeister, E., 13, 48

Cagan, P., 13, 58, 327Carlin, W., 8, 377Chaikin, S.E., 56Chiarella, C., 11, 23, 30, 31, 46, 55, 56, 61, 65,

66, 67, 73, 77, 89, 127, 129, 130, 132,159, 162, 171, 174, 177, 181, 189, 262,264, 265, 293, 294, 315, 336, 339, 343,376, 377, 379, 381, 382

Chua, L.O., 66Cripps, R, 379Cross, R.R., 342, 347

Dixon, H.D., 378Dobell, R., 13Dore, M., 373, 374, 375, 376Dornbusch, R., 58, 61, 336

Dutt, A., 189, 373, 378

Eichenbaum, M., 64, 338, 382Epstein, G., 188, 378

Fair, R., 180, 297, 342, 379Ferri, P., 127, 142, 143, 144Fischer, S., 8, 14, 15, 31, 32, 36, 38Flaschel, P., 10, 11, 16, 23, 31, 37, 45, 46, 56,

65, 67, 73, 77, 92, 114, 127, 129, 130,132, 140, 141, 146, 160, 162, 171, 174,177, 181, 189, 191, 238, 257, 258, 262,264, 265, 272, 274, 293, 294, 315, 336,339, 343, 373, 376, 377, 378, 379, 381,382

Flood, D., 294Franke, R., 10, 16, 45, 63, 140, 146, 181, 269,

273, 294, 311, 378Frankel, J.A., 58Franz, W., 340, 353Froot, K., 58Fujino, S., 14

Galbraith, J.K., 342Gale, D., 30Gandolfo, G., 67Gantmacher, F.R., 66George, D.A.R., 53Gintis, H., 188, 378Godley, W., 379Goldman, S.M., 13Gong, G., 132, 238, 382Goodwin, R.M., 13, 15, 35, 97, 127, 190, 289,

291, 307, 374, 376Gordon, R., 342Grandville, O., 199, 294Greenberg, E., 35, 127, 142, 143, 144Groh, G., 11, 23, 65, 73, 77, 129, 130, 132,

140, 177, 181, 189, 262, 264, 265, 315,

394

336, 339, 376, 377, 378, 379, 381, 382Groth, C., 60, 87, 162Guckenheimer, J., 66, 67

Hadjimichalakis, M.G., 13, 29, 83Hahn, F., 11Hairier, E., 66Hall, S.T., 64, 338, 342Hayakawa, H., 13, 30, 83Heap, H.S., 347Henin, P.Y., 18, 21, 379Hirsch, M.W., 66, 141, 152, 255Holmes, P., 66, 67Hommes, C.H., 61Howitt, P., 378

Ito, T., 127, 143Iwai, K., 14

Jackman, R., 377Johnson, H.G., 13, 25Jones, H.G., 236

Kahn, C.M., 48Kaldor, N., 149, 276, 279, 280, 291, 293, 307,

374, 375, 376Karakitsos, E., 8Katz, L.F., 342Kearney, C., 48Keynes, J.M., 16, 268, 272, 315, 334Khibnik, A., 67Khomin, A., 61, 67, 382Klein, L., 379Klundert, T., 342Koper, C., 11, 65, 73, 77, 129, 130, 132, 264,

336, 339, 376, 377, 379, 381Kuh, E., 142, 347, 348Kuznetsov, Y.A., 87

Laxton, D., 327Layard, R., 377Leslie, D., 8Lindbeck, A., 343Lorenz, H.W., 30, 67Lowe, P., 294Lux, T., 63, 181, 294, 302, 311

MacDonald, R., 48Malinvaud, E., 18Malliaris, A.G., 66, 141, 148, 156Mankiw, G., 338Marglin, S., 260, 346, 373Marwah, K., 379Marx, K., 348, 355Matsumoto, A., 294

McCallum, B.T., 39, 48Medio, A., 67Metzler, L.A., 181, 279, 375, 376Michel, P., 18, 21Miyao, T., 13, 29, 87, 93, 99, 120, 158, 214,

250Murphy, C, 261, 262, 335, 336, 337, 377, 379Murphy, K.M., 342

Nagatani, K., 13, 14, 28Nickell, S., 377Norsett, S.P., 66

Okuguchi, K., 13, 83Orphanides, A., 1, 8, 11, 13, 25, 28, 30, 112,

172, 272, 375Oswald, A.J., 348Oxley, L.T., 53

Papell, D., 58Parker, T.S., 66Patinkin, D., 163Paul G., 342Perko, M.L., 66Phelps, E., 342Phillips, A.W., 347Place, C.M., 66Pohjola, M., 143Powell, A., 261, 262, 335, 336, 337, 377, 379Puu, T., 67

Rankin, N., 378Rogerson, R., 342Romer, D., 8Rose, H., 14, 15, 18, 32, 33, 34, 35, 36, 37, 38,

95, 98, 99, 127, 146, 147, 148, 153, 155,199, 200, 290, 291, 307, 327, 375, 376

Rowthorn, B., 378

Sala-I-Martin, X., 378Samuelson P., 39, 180Sargent, T., 4, 5, 10, 11, 13, 16, 20, 24, 26, 31,

37, 39, 40, 41, 43, 44, 45, 46, 47, 61, 70,75, 81, 91, 103, 104, 110, 132, 136, 140,161, 162, 163, 164, 229, 243, 260, 261,270, 272, 273, 285, 376

Schaik, A., 342Semmler, W., 10, 11, 16, 23, 65, 73, 77, 129,

130, 132, 140, 146, 177, 181, 189, 238,262, 264, 265, 315, 336, 339, 376, 377,378, 379, 381, 382

Sethi, R., 31, 32, 37, 56, 61, 92, 140, 146, 148,162, 257, 258

Sidrauski, M., 13, 28Sijben, J., 13, 14, 24, 29

395Author index

Skott, P., 14, 284, 375, 378Smale, S., 66, 141, 148, 152, 255Snowdon, B., 375, 376, 377Snower, D., 343Solow, R., 1, 8, 11, 13, 24, 25, 28, 30, 32, 39,

64, 95, 112, 140, 172, 180, 181, 272, 338,375, 382

Soskice, D., 8, 377Stein, J., 10, 14, 31, 32, 37Stiglitz, J.E., 95, 142, 143, 145, 148, 342Straiger, D., 342Strogatz, S.H., 66, 67, 323Summers, L., 342

Tambakis, D., 327Taylor, L., 58, 64, 189, 338, 378, 382Tobin, J., 6, 13, 24, 30, 69, 87, 276, 280, 283,

287, 290, 291, 376

Topel, R., 342Tsiang, S.C., 48Tu, P., 67Turnovsky, S.T., 4, 5, 8, 10, 16, 24, 45, 47, 48,

50, 81, 159, 164, 375, 376, 380

Vane, H., 375, 376, 377Vitt, A.A., 56

Wallace, N., 31, 43, 44, 47, 81, 91, 161, 162Wanner, G., 66Westphal, U., 379Whitley, J., 379Wiggins, S., 66, 67, 306Wolfsetter, E., 114Wynarczyk, P., 375, 376, 377

Zoega, G., 342

396 Author index

Subject index

accelerator, 36, 38, 234accumulationasset, 26capital, 15, 16, 18, 27, 28, 75, 133, 144,146, 171, 342, 355

government debt, 182, 286AD—AS growth model(s)the bastard limit case, 268—275classical, 39Keynesian, 16Keynes—Wicksell variant, 230

adaptive expectations, 16—20, 39—46, 48—53,237, 290, 297

backward looking, see backward lookingand Benassy business cycle model, 232and Cagan model, 58forward looking, see forward lookingand myopic perfect foresight, 47, 49, 50,55, 109, 159

and pure monetary cycle, 159—160, 162,215

and Sargent model, 31, 45and Tobin model, 28—31, 81, 109,243—245, 253

adjustment mechanismmacroeconomic, 348Metzlerian inventory, 22, 63, 181Phillips-type, 161, 216price, see price adjustmentrate of employment, 229, 341, 353, 358wage, see wage adjustment

agentsunder adaptive expectations, 48with complete knowledge, 54with optimizing behavior, 13, 30

aggregate demandand bastard limit case: AD—AS growth,269, 271

and Benassy business cycle model, 235

and IS equilibria, 196—197and Kaldor—Tobin model, 285, 290,346—347, 352

and Keynes—Metzler model, 296—298,317, 322

and Keynes—Wicksell model, 133, 171and Keynesian cyclical growth modelwith infinite speed of price adjustment,217

and Keynesian monetary growth model,179, 181, 307

animal spirits, 341, 362asset market equilibrium, 21, 239and Kaldor—Tobin model, 285and Keynes—Metzler model, 295, 299and Keynes—Wicksell model, 129, 131and Keynesian cyclical growth modelwith infinite speed of price adjustment,217

and Keynesian monetary growth model,178, 180, 263, 270

and Tobin model, 72, 83, 93, 104, 113

backward lookingcomponents of delayed adjustments ofinflationary expectations, 376expectations, 43, 159, 162, 216inflationary expectations, 76, 80mechanism, 48see also expectations

bifurcationdegenerate, 87, 301, 320and Keynes—Wicksell prototype model,157—158, 160—161

and steady-state inflation and complexdynamics, 332—333

subcritical, 87, 301, 309, 320supercritical, 87, 301—304, 306, 309, 320see also Hopf bifurcation

397

bond market, 102, 106, 120boundedness, 66, 123mathematical, 324, 325

budget restrictionsgovernment, 26—27, 71, 74, 106, 110, 128,314

of households, 26, 71, 314intertemporal government, 110

business cycles, 18, 127Benassy model, 231—235Goodwin, 35,implications of, 190—199Kaldorian approach of, 284Rose, see Rose employment cyclemacrodynamics of, 373—374see also growth cycle

Caganeffect, 69, 102, 253model, 47—55, 159, 162

capacity utilization, 18, 229, 285, 377desired rate of, 20, 180, 345natural rate of, 176normal rate of, 20, 284, 345, 380rate of: and AS—AD model, 62; andBenassy business cycle model, 231—233;and Keynesian IS—LM growth model,261—263; and Keynes—Metzlermonetary growth model, 316; andKeynesian prototype monetary growthmodel, 19; and Keynes—Wicksellmonetary growth model, 171, 175—180,279; and neo-Keynesian monetarygrowth model, 17, 325; andnonlinearity and real cycle, 226, 229

steady-state rate of, 182capital accumulation, 21, 342, 355, 373and Keynes—Wicksell model, 15, 171and neoclassical disequilibrium, 144and Neo-Keynesian monetary growthmodel, 18

and Tobin model, 27—28, 123capital depreciation, 134capital stock, 69, 262actual rate of growth of, 237, 361and ceiling and floor for economicfluctuations, 326—327

and classical regime, 345growth equation for, 111growth rate of, 92, 110, 118, 143, 237and inventories, 134and Kaldor’s trade cycle, 291and labor demand, 74and macroeconomic magnitudes, 72natural rate of growth of, 346

normal rate of utilization of, 345and private and government savings, 76Solovian growth of, 16trend growth rate of, 342, 365under-/overutilized, 179, 230utilization of, 18, 24, 62, 164, 175,179—180, 216, 269, 274, 316, 376

ceiling(s)absolute full employment, 21, 94,143—144, 234—235

to economic activity, 144, 264, 324—325to validity of the Goodwin growth cycle,145

central bank, 237, 381chaos, 143, 282, 333chartists, 59Classical growth cycle (Goodwin model), 15,

35, 127, 307Classical regime(s), 17—18, 262, 338, 345consumption, 28, 134, 197aggregate planned, 73function, 18, 26, 41, 115, 137, 337

corridoraround steady-state, 149, 193of stability, 287, 301, 303—304, 312

cost push, 95, 236, 263effect, 152, 275

cross-dualgrowth cycle dynamics, 141nature, 289

cycles: see business cycles, growth cycles

damped fluctuations, 368debtfinance, 105, 380government, 69—70, 72, 102, 105, 117, 136,182, 381

deflationthe case of no nominal wage, 325—328implied wage, 329nominal wage, 315, 375price of, 55, 328

degenerate Hopf bifurcation, 87, 301, 320and Cagan and Tobin stability condition,87

and general Keynes—Metzler monetarygrowth model, 320

and Keynes—Metzler model, 301demandaggregate, see aggregate demandasset, 55, 102, 105, 180bond, 105excess, see excess demandexpected, 197—198, 346gap, 33, 76, 197

398 Subject index

law of, 134, 261stock, 73, 75, 132pressure, 14, 18, 37, 70, 95, 128, 171, 181,263, 314, 376

pull, 62, 135, 277, 380derivative control, 348deposits, 39, 104, see also savingsdesired rate of capacity utilization, 237, 279discrete timeanalysis, 381dynamical models, 65, 67, 129, 143

disequilibriumgoods market, see goods market

disequilibrium,IS-, 19, 20, 22, 179, 279, 283, 285—286and Kaldor—Tobin model, 283—290and Keynes—Metzler model, 22—24,293—295, 314

and Keynes—Wicksell model, 14, 34—35,129—136, 172, 179

and Keynesian monetary growthdynamics, 19—21, 24, 172

labor market, 64, 92—94, 147, 161, 193,216, 290, 335

macrodynamics, 12, 17money market, 69, 82—84, 112, 125,247—250

neoclassical approach, 143neo-Keynesian analysis, 17, 20and Rose employment cycle, 147and versions of Tobin model, 14, 27—30,69, 112—114, 124, 173

distributional conflict, 378dividend(s), 134, 296dynamic multiplierof integrated Keynes—Metzler model,293—297

of Kaldor—Tobin model, 283—293of Keynesian model, 15, 21, 62, 63, 197,199, 276

dynamically endogenous variables, 45,175, 265, 269, 273

effective demandand AS—AD growth model, 16and Benassy business cycle model,232—234

and IS—LM growth model, 261and Kaldor—Tobin model, 343, 345and Keynes—Metzler model, 375, 308and Keynes—Wicksell model, 14, 187—189,230

and Keynesian monetary growthdynamics, 19—20, 173, 175—176, 180,193, 375, 378

efficiency units, 240elasticityof money demand, 141, 155of substitution, 146, 255

employmentadjustment of the rate of, 65, 229, 341,345, 353, 358

ceiling(s) to, see ceiling(s)cycle extensions, 199—207endogenous long-run, 340—341, 343, 372,376—377

full, see full employmentand Goodwin cycle, 141, 143and Goodwin—Rose cycle, 291and Keynes—Metzler model, 316, 319,325

and Keynes—Wicksell cycle, 34, 36, 253,271

and Keynesian IS—LM growth model,240, 260, 263—264

and Keynesian monetary growth model,19, 176, 180, 279

NAIRU-type rate of, see NAIRUemployment ratenatural rate of, 20, 45, 64, 94, 143—144,176, 194, 337, 342, 350, 353, 357, 370,372—373

over- and under-, 20, 171, 230, 371, 377rate of, 229, 325, 336, 347, 362,Rose cycle, see Rose employment cyclesteady state rate of, 143—144, 280, 340,370

endogenous animal spirits, 364—365endogenous business cycle approach, 374endogenous determinationof depreciation rate, 326of natural growth and NAIRU rate ofemployment, 23, 65, 343, 345, 356—358,365—368, 370—374

long-run growth and employment,341—348, 372, 376—377

of nominal wages, 268of steady state rate of growth andemployment, 280, 340—341

of taxes per unit of capital, 286endogenous money supply rule, 33endogenous trend term of investment and of

labor force growth, 377endogenous variables, 45, 48, 91, 182, 268,

340dynamically, 41, 175, 265, 269, 273statically, 41, 175, 265, 269, 273, 347

equityfinancing, 132—133supply of firms, 296

399Subject index

equilibriumasset market, see asset marketequilibrium

full employment, 77, 106—107, 146—147,243, 274

general, 13, 17, 30, 70, 81, 102—112, 243,246, 252

goods market, see goods marketequilibrium

IS—, 18—19, 179, 196—197, 199IS—LM, see IS—LM equilibriumand jump-variable technique, 50—51, 55money market, 77, 83, 124, 129, 159, 244,247, 299, 337, 345

rate of inflation, 59, 83steady state, 15, 60temporary, see temporary equilibriumand Tobin models, 13, 71—82

Euler’s theorem, 243excess demand, 34, 107, 153, 158, 167,

196—197, 286, 328excess supply, 83, 88, 193, 197exogenous desired rate of capacity

utilization, 80exogenous labor supply, 41exogenous natural rates, 350—351exogenous shocks, 346exogenous trend growth, 361exogenous variables, 94, 317, 353expectation(s)adaptive, see adaptive expectationsasymptotically rational, 61, 80, 129, 140and augmented money wage Phillipcurve, 16

backward looking, 43, 76, 80, 159, 162,216

demand, 234, 296—297forward looking, 60, 216, 278, 376heterogeneous, 58—61income, 76, 296inflationary, see inflationary expectationsmodeling of, 47—61myopic perfect foresight, 30, 41—43, 47,50, 58, 104, 109, 159, 161, 216, 246—247

p-star, 235, 237, 273, 278, 335, 376and pure monetary cycle, 159—168, 2 15rational, 13, 46—48, 52—53, 56, 58—59, 126,336—337

regressive, 61, 72, 104, 159, 160, 190,215

sales, 10, 22, 63, 276, 279, 293, 296—297,302, 309, 314, 321, 337, 376

expected rate of inflation, 26, 28, 29, 36, 42,49, 54, 60, 71, 73, 76, 79, 80, 94, 119,163, 174, 184, 215, 316

feedback loops in labor market adjustmentprocesses, 341

financial asset, 15financial fragility, 104financial markets, 64, 378fixed proportions technology, 74, 128, 147,

235, 237, 260—261, 268, 275, 316flexibilityin employment decision of firms, 345in interest rates, 99, 129, 154, 158, 174,199, 207, 209, 214, 258, 313

in nominal rate of interest, 154—155in price, 33—34, 40, 102, 122, 158, 199—200,204, 206, 211, 268—269, 272—274,306—307, 313

in wages, see wage flexibilityforecast, 238, 380errors, 48

forward lookingClassical monetary growth model, 43expectations, 52, 60, 76, 162, 216, 278, 376and fundamentalists, 59p-star expectations, 278, 376see also expectations

full employmentabsolute, 21, 94, 143barrier, 327equilibrium, 77, 106, 146—147, 243, 274exogeneity of, 280, 340; and AS—ADmodel, 274; and neoclassical model, 27,40; and Keynesian model, 44; and Say’sLaw, 72, 74, 83, 104; and Solow model,147; and steady-state model, 39, 25 1;and Tobin models, 72, 74, 83, 104,242—243

full employment ceiling, 21, 94, 143—144,234—235

full employment labor intensity, 34, 251, 274fundamentalists, 59

goods demandand AS—AD growth model, 16and Kaldor—Tobin model, 343—345, 347and Keynes—Wicksell model, 34, 125, 134,335

and Keynesian monetary growthdynamics, 19, 21, 173, 176

and Rose employment cycle, 153and Tobin model, 76

goods market disequilibriumand bastard limit case: AD—AS growth,271—272

and Kaldor—Tobin model, 283, 285, 341,344

and Keynes—Metzler model, 64, 293—296,

400 Subject index

314, 375and Keynes—Wicksell model, 129—13 1,135—136, 335

and Keynesian monetary growthdynamics, 19, 24, 171—179

and Rose employment cycle, 147and pure monetary cycle, 161

goods market equilibrium, 14, 21, 123, 239and AD—AS growth, 41, 270and Keynes—Metzler model, 297and Keynes—Metzler model with realwage and accumulation dynamics,304—305, 307, 314, 337

and Keynes—Wicksell model, 164,171—172

and Keynesian monetary growthdynamics, 63, 178, 180—181, 195, 197,218, 222—223, 235, 260, 263, 279

and Say’s Law, 113and Tobin model, 27

Goodwin model, 15, 35, 142, 144, 146, 148,149, 154, 184, 255

Goodwin classical growth cycle, 15, 35, 127,307

Goodwin growth cycle, 3 5—36, 129,140—146, 153, 164, 190, 255

Goodwin/Rose growth cycle, 102, 154, 291,307, 313

governmentasset, 104budget constraints, 26—27, 71, 74, 106,110, 128, 314

debt, 69—70, 72, 102, 105, 117, 136, 182,381

debt accumulation, 182, 286, 297deficit, 111expenditure, 72, 74, 103, 105, 110—111,115, 117, 120, 137, 155, 298, 337

monetary and fiscal authority, 83, 93, 103,113, 130, 177, 239, 263, 270, 284, 295,344

savings, 74, 76, 105government budget restraint (GBR), 74, 106,

110, 114, 128, 294, 304growthfundamental equation of, 25monetary, see monetary growthnatural rate of, 64, 75, 105, 117, 139, 236,311, 337, 342, 346, 355—357, 366,370—373

real, see real growthgrowth cyclesRose, 91, 92, 129, 146—154, 200, 207Goodwin, 35, 92, 121, 129, 140—146, 186,206, 289—290

Goodwin/Rose, 155—158monetary, 90, 154—158pure monetary, 167—168, 216—220

Harrod—neutral technical process, 235, 239,241, 335, 377

Hopf bifurcationand Cagan and Tobin stability effect, 102and Cagan and Tobin stabilityconditions, 87

and employment cycle extensions,200—201

and general Keynes—Metzler monetarygrowth model, 319—323

and inventory cycle, 309—312and Kaldor—Tobin monetary growthmodel, 288—289, 293

and Keynes—Metzler model, 300—306and Keynesian monetary growth model,212—214

and pure monetary cycle, 218and Tobin model, 29, 86—87, 99, 249—251

hyperinflation, 58hysteresis, 280—281, 340, 346—347, 353,

355—357, 363, 365

imbalance(s)in goods market, 15, 62, 134, 274, 279, 286in labor market, 19, 61, 274in utilization of capital stock and laborforce, 274

incomedisposable (perceived), 25—27, 30, 40, 44,70, 72—76, 100, 103, 105, 11, 121, 129,131, 380; Barro definition of, 110, 118;and Tobin effect, 102

distribution of, 14—15, 27, 143, 184, 241,252, 307, 373, 378, 382

expectations, 271, 296wage, 236, 238, 241

incompletely specified economic models, 42,45, 62

inflationactual rate of, 26, 81, 115, 137, 159, 216,237, 335

and Benassy model, 18, 231and bastard limit case: AD—AS growth,270

and Cagan model, 13, 54expected rate of, see expected rate ofinflation and instability, 88, 102

and IS—LM model, 17, 19, 62, 128, 172and Kaldor—Tobin model, 291and Keynes—Metzler model, 22, 309, 316,328

401Subject index

and Keynes—Wicksell model, 14, 15, 35,39, 131, 135, 164, 171—172, 178—179

medium run, 94, 162, 237, 275and Mundell effect, 119and Phillips wage-curve, 40, 315price, 49, 99, 129, 136, 152, 365, see alsoprice and wage inflation and p-star 59,235, 237

and pure monetary cycle, 160, 162—163rate of labor utilization, non-accelerating,342

repressed, 17, 18, 20, 94, 181, 338, 345and Rose (cycle) model, 33, 35, 36, 38, 152steady state of the rate of, 38, 42, 83, 94,215, 251, 309, 329, 330, 331, 359, 361,363

and Tobin model, 27, 71, 73, 79—80,83—84, 94

wage, 33, 38, 62, 315, see also price andwage inflation

inflationary expectations, 13and AS—AD growth model, 61and Classical model, 41and Goodwin growth cycle, 121and jump-variable technique, 49, 91and instability, 102, 227and Kaldor—Tobin model, 291and Keynes—Wicksell growth model,36—39

and Keynes—Metzler model, 302, 304,309, 321

and Keynes—Mundell effect, 185and Keynesian AS—AD growth model, 16and Keynesian monetary growth model,376

medium run, 38, 95, 174, 216, 227, 236,328

and monetary growth cycle, 155, 159, 160and p-star concept, 236—238and pure monetary cycle, 129relaxation oscillation in, 57and Tobin model, 28, 30, 76, 79—80, 83,89, 94

information, 52, 56, 214, 306, 322incomplete, 17long run, 163

instabilityand adaptive expectations, 29, 3 1, 159,216, 245

and Benassy model, 197, 233—235and Cagan effect, 69, 102complete, 43global, 81and Goodwin model, 97, 154and Keynes—Wicksell model, 43, 258

and Keynesian model, 63, 195—196, 198,201—202, 206, 212, 215, 225

local, 31, 35, 69, 81, 86, 154, 156, 168, 215,234, 250, 258, 309, 358

and myopic perfect foresight, 31, 5 1, 81,84, 159

and Rose cycle, 35, 148, 156, 168and Rose effect, 99, 122, 201, 203saddlepoint, 28, 31, 47, 162and Tobin models, 29, 31, 69, 82, 86—87,90, 120, 245, 249—250, 252—253

interest elasticity of money demand,140—142, 155

interest payments, 104, 105, 128, 181interest rate, 36, 104, 174, 271, 273, 313, 316,

380and Benassy model, 232and Cagan model, 159and Goodwin/Rose growth cycle, 154flexibility of, 99, 129, 154, 15 8, 174, 199,207, 214, 258, 313

and Keynes—Metzler model, 316, 319nominal, 15, 26, 41, 104—106, 134,154—155, 159, 163, 167, 178, 184, 208,217, 220, 269, 271, 325

and pure monetary cycle, 167and Rose cycle, 32, 129real 36, 37, 41, 104, 128, 132, 345

interest rate effects, 185inventorychanges, 134—136, 178, 261, 286, 294, 296,314

dynamics, 309, 311, 321investment, 297, 325Metzlerian adjustment process, 22—23, 63,181, 199, 276, 279, 283, 286, 294, 302,315, 337, 341, 375—376

policy, 199, 316, 345investment behaviorand Benassy model, 199, 234and Classical model, 40, 44independent, 112, 123, 133and Kaldor—Tobin (endogenous growth)model, 341, 343, 355—357, 361, 377

and Keynes—Wicksell model, 18, 31,124—125, 128—129

and Keynesian growth, 19, 176, 180, 263and Keynesian prototype model: IS—LMgrowth, 62, 185

and pure monetary model, 165and Rose cycle, 33—34, 149and Tobin model, 14, 71

investment demand, 76, 135, 197, 308, 337,346

investment decision(s)

402 Subject index

and Benassy model, 232long-run, 355and Kaldor—Tobin model, 345and Keynes—Wicksell model, 14—15, 37,39

investment equation, 182, 355investment and finance, 134—135, 296, 380investment functionand bastard limit case: AD—AS growth,269, 271

and Benassy model, 235and inflationary expectations, 37, 236and integration of monetary and realcycle, 169, 174, 227

and Kaldor trade cycle, 291and Keynes—Wicksell model, 36, 131—133and Keynesian monetary growth, 19and monetary cycle, 169, 174, 218, 220and neo-Keynesian monetary growth, 18nonlinearity in, 152, 159, 166, 169, 215,220—222, 226—228, 256, 282, 307, 309,375

and real cycle, 169, 174, 215, 221, 226slope of, 166and Tobin model, 25

investment per unit of capital, 132, 190investment-savings relationship, 151IS-curves, 199IS-(dis)equilibrium, 22, 279, 171, 175, 179and Benassy cycle, 199and Keynes—Metzler model, 279, 283,285—286

and Keynes—Wicksell model, 283Keynesian, 18—19, 195—199

IS—LM equation(s), 164, 232, 269, 271, 273IS—LM equilibrium, 62and Benassy cycle, 197—199, 231to goods market disequilibrium, 291, 314and the Keynes—Wicksell model,164—165, 271

Keynesian, 19, 21, 128, 178, 260, 267,271—272, 274, 276, 278, 283

and Keynesian AS—AD model, 16, 19, 229IS—LM growth, 164, 172, 176, 182and Benassy cycle, 230and the Keynes—Wicksell model, 164Keynesian, 20—21, 62, 23 8, 259—268,274—275

IS—LM subsector of Keynesian monetarygrowth, 184—189

jump(s)in inflationary expectations, 81, 91—92,160

in linearization of the stable manifold, 53

in price level, 43, 46, 53, 91in wage level, 46

jump-variable technique, 31, 46—59, 81,91—92, 162, 199

Kaldordynamic multiplier process, 375—Tobin model, 280, 283—293, 341,343—348, 350—352, 356—358

trade cycle, 276, 280, 283, 291, 307, 374Keynes effect, 154, 184, 207, 210, 218, 235,

293, 318, 321Keynes—Goodwin model, 15, 190—199—Marx model, 248—Metzler model, 10, 22—24, 62—64,278—339, 341, 374; integrated, 293—314,342; simplified, 315, 317

—Wicksell model, 11, 14—15, 18, 19, 22, 24,31—39, 45, 62, 64, 70, 99, 112, 125,127—172, 174, 182, 215, 230, 279;prototype, 129—136; real cycle of,152—154, 179; smooth factorsubstitution, 242—243, 253—259

Keynesian cyclical growth, 214—229investment nonlinearity and real cycle,221—226

pure monetary cycle, 217—220real and monetary cycles, 226—229

Keynesian (dis)equilibriumIS—, 17, 19, 22, 44, 128, 165, 178, 283IS—LM, 19, 22, 44, 128, 165, 178, 283LM—, 15, 17, 19, 22, 44, 128, 135, 165, 178,283

Keynesian IS—LM growth, 184—189basic prototype, 19—20, 21, 23, 61—62, 172generalized, 259—268

Keynesian model of monetary growth, 10,11, 125, 175—184, 207—214, 374—380

AS—AD, 16—17, 19, 44—46, 62, 163, 174,230, 268—275

and Classical model, 39—44and the expectation mechanisms, 215—217and Goodwin growth cycle, 140—146neo-, 17—18, 19—21and Rose employment cycle, 127, 146—154and smooth factor substitution, 242—243,259—275

Keynesian regime, 17, 19, 20, 22, 143, 177,351

labordemand, 18, 34, 74, 148—149, 159, 353market disequilibrium, seedisequilibrium, labor market

403Subject index

productivity, 142, 235—237, 241, 327supply, 16, 18, 41, 74—75, 144—145, 153,171, 240, 337, 342, 343, 345, 346

utilization of, 17—19, 24, 33, 62, 179, 180,274, 316, 326, 342, 347

law of demand, 134Liapunov functions, 66—67, 87, 129, 141,

145—148, 150, 254—255, 333liquidity preference, 269liquidity trap, 185, 190, 304limit cycles, 330—33 1; see also business

cycle, growth cycleand Keynes—Metzler models, 301—304,308—309, 320—321, 323

and Tobin model, 69, 86, 91—92, 123, 250limit limit cycles, 56—5 8, 66, 90—91LM-curve, 199LM-equation(s), 26, 37, 131LM-equilibrium, Keynesian, 15, 135loans, 380, 382locally asymptotically stable dynamics,

99—101, 119, 156, 194, 211, 227, 249,290—291, 293, 306, 308, 311, 318, 321,332, 341—342, 351, 354—355

locally unstable steady-state dynamics, 15,34, 88, 148, 169, 204—205

long-run employment and growth(endogenous), 340—348, 359, 368, 372,376—377

long-run inflation, 76long-run integrated macrodynamics of, 64,

189, 338lump-sum taxes, 75, 298

marginal product of labor, 179, 256marginal productivityprinciple, 269, 271rule, 15, 32, 46, 231, 243, 253, 261, 263,269, 273, 326, 337

marginal propensity to consume, 197, 241marginal wage cost(s), 16, 229, 261, 268, 272,

326, 377rule, 175, 231

market adjustments, 221goods, 281, 283, 291, 341labor, 341, 355, 376pure money, 159

markup pricing, 229, 232partial, 236static, 95, 181

Marxian view, 356neo-, 378

monetary growth models,Classical, 43, 374disequilibrium, 12—18, 20, 90, 92, 112

Kaldor—Tobin, 283—293, 3 76Keynes—Metzler, 22, 62—64, 279;integrated, 293—314; six-dimentional,314—335

Keynes—Wicksell, 14—15, 31—39, 64, 112,127—172, 230, 253—259, 378

Keynesian, 10—11, 16, 19—24, 76, 112, 134,163, 171—172, 173—241, 33 5, 341,374—380; IS—LM, 61—62, 165, 259—275,278; missing prototype, 173—241;prototype, 22, 62, 128, 340, 377

neo-classical, 13, 32, 112, 123, 128Tobin, 12, 14, 24—31, 37, 69—126, 132, 230,243—253

monetary shock, 368moneydemand, see money demandmarket disequilibrium, 69, 82—84, 112,125, 247—250

market equilibrium, 77, 83, 124, 129, 159,244, 247, 299, 337, 345

quantity theory of, 30, 59, 75, 237—238superneutrality of, see superneutrality ofmoney supply, 13, 24, 27—28, 39, 42, 75,80—81, 103, 117, 139, 163, 185, 236, 251,337, 362

wage, see Phillips curve, money-wage;wage adjustments, money

wage rigidity, 272money demandand Cagan effect, 69interest elasticity of, 141, 155and jump-variable technique, 49, 55—56and Goodwin growth cycle, 140,and Kaldor—Tobin monetary growthmodel, 291, 293

and Keynes—Metzler model, 316, 326and Keynes—Wicksell model, 134,300—306

and Keynesian monetary growth model,208, 212

and Tobin effect, 250—251and Tobin model, 73, 80—86, 88—90, 97,105

monopolistic competition, 17, 32, 379multiplier, dynamic, see dynamic multipliermultiplier instabilities, 63, 198Mundell effect, 119, 159, 169, 174, 184—185,

208, 215, 218, 234normal, 215, 218, 308

Murphy model (for the Australianeconomy), 261, 335—338, 379

myopic perfect foresightand adaptive expectations, see adaptiveexpectations

404 Subject index

and AD—AS growth (bastard limit case),273

and Cagan model, 58, 159and Classical model, 41—43and Goodwin growth model, 36, 164and heterogeneous expectations, 61and jump-variable technique, 47—56and Keynes—Wicksell model, 38and Keynesian cyclical growth, 196, 216and Keynesian growth model, 46and pure monetary cycle, 161—163and Tobin model, 26—31, 36, 80—81, 91,246

myopic perfect foresight limit, 50, 51, 56,59, 91, 162

NAIRU employment rate, 19, 23, 94, 180,264, 280, 326, 329, 342—349, 350, 353,356, 365, 368, 371

natural rate of economic growth, 23, 280,340

natural rate hypothesis, 342neoclassical capital accumulation, 15neoclassical monetary growth, 13, 32, 112,

123, 128neoclassical production function, 179, 229,

230, 242—243, 256, 260, 262, 266, 268,337

neoclassical synthesis, 40, 42, 163, 272—273neo-Keynesian monetary growth, 17—21,

261—262, 376nonlinearitiesinvestment, 152, 159, 166, 169, 215,220—222, 226—228, 256, 282, 307, 309,375

natural, 65, 73, 148—149, 227, 282, 299,319, 322, 325, 350

and viability, 123, 149, 169, 282, 315, 358and wage adjustment, 202—204, 282, 319

non-Walrasiandisequilibrium analysis, 17equilibrium analysis, 377model, 231regime switching analysis, 326theory, 181

normal employment, 326, 347normal labor supply, 74, 171normal rate of capacity utilization, see

capacity utilization

Okun’s Law, 180, 216, 229open economy, 56, 335—336, 377, 379, 381open market policies, 103optimizing behavior, 13, 30intertemporal, 126

optimizing macroeconomic frameworks, 28,30

orbitclosed (Goodwin), 36, 121, 141, 144,148—152, 154, 201, 255

closed (Hopf), 218see also business cycle

oscillators, 14, 170, 304relaxation, 57, 58, 66, 90—91

output—capital ratio, 217, 221, 235, 265, 273overshootingof exchange rates, 336mechanism (Goodwin), 36, 92, 140, 142,184, 190

overtime work, 74, 94, 327, 337

perfect competition, 33perfect foresight, see myopic perfect

foresightpersistent cyclical motions, 34, 287, 368persistent fluctuations, 15, 91—92, 169, 282,

332, 360, 375Phillips curve, 10, 14, 16, 39, 45, 345in Benassy model, 231—233expectations augmented, 16, 18, 314in Goodwin model, 142, 144kink in, 144, 282, 332, 335, 359, 362, 370,375

Kuh-component of, 142, 348loop, 153money-wage, 14, 16, 20, 23, 36, 127, 229,232, 256, 268, 273, 297, 327—328, 332,336, 347, 358, 369—371

and nonlinearities, 127, 148, 152, 202—203,256, 282, 307, 358

price, 180—181, 297, 314, 380in real and monetary cycle, 169—170real wage, 36, 46, 127, 273in Rose cycle, 34, 36—37, 148, 152—153wage, 314, 353wage inflation, 40, 315

Pigou effect, 234, 235Poincare—Bendixson theorem, 66, 204, 206,

250and Benassy model, 231and Kaldor model, 287and Metzlerian model, 311and pure monetary cycle, 166—168, 218,220

and pure real cycle, 225and Rose growth cycle, 34, 129, 152—153,257, 307

and Tobin-type models, 89policy rules, 105, 114, 380, 381population growth, 74

405Subject index

portfolio, 75, 106, 132, 135adjustments of (in money market), 48, 55choice mechanism, 25, 30

price adjustmentand bastard limit case: AS—AD model,269

and Cagan monetary growth model, 49infinite speed of, 160, 217and Keynes-Metzler monetary growthmodel, 22

and Keynes—Tobin monetary growthmodel, 287, 328

and Keynes—Wicksell model, 131, 134,179, 202

and Keynesian monetary growth model,19, 207—208

and Phillips curve, 181and Rose model, 127and Tobin model, 95

price flexibility, 33—34, 40, 102, 122, 158,199—200, 204, 206, 211, 269—269,272—274, 306—307, 313

price and wage inflation, 46—47, 94—95,180, 229, 231, 236—237, 275, 279, 376

production, 18, 159, 337, 377, 380and AD—AS growth (limit case), 269, 271and Benassy cycle, 231and Goodwin model, 143and Kaldon—Tobin models, 284—285, 343,352

and Keynes—Metzler models, 294,296—297, 326

and Keynes—Wicksell models, 130,132—135, 163, 242, 253, 255, 259

and Keynesian IS—LM model, 239,259—260, 262, 265—266

and Keynesian monetary growth, 19, 45,176—179, 181, 242

and myopic perfect foresight, 216and Rose model, 32, 256and Tobin models, 30, 69, 71—73, 76, 83,93, 103, 113, 242—243, 245—246, 251

profit, 18, 105, 296, 327actual rate of, 72expected, 132, 296income after taxes, 74—75, 152led, 188rate of, 15, 26, 33, 37, 73, 75, 84, 106, 139,186, 226, 243, 273, 305, 319, 346

profit-maximizing, 262, 271full capacity, 260—261marginal productivity, 15, 271productive capacity, 261

profit sqeeze mechanism, 35, 146, 190full employment, 189

overshooting, 140profitability effect of real wage, 15, 18, 186,

305p-star expectations (forward looking),

235—236, 335, 376p-star inflation rate theory, 59

quantity adjustment (process), 10, 22, 63, 64,279, 281, 285—286, 294, 341, 362, 380

delayed, 287, 299Kaldor, 279Metzier, 279sluggish, 278, 283

quantity constraints, 315, 326quantity theory of money, 30, 59, 75, 237,

238

rate of interest, see interest raterate of profitequilibrium value of, 79, 85and investment, 33, 221, 262, 345, 377see also profit

real balances (per capital), 27—28, 46, 73, 79,88, 121, 165, 273, 308

real capital accumulation, 16, 75real cycle, 169—171, 205, 207, 221—229, 379Goodwin, 97, 121Goodwin/Rose, 154Kaldor (trade), 283in Keynes—Wicksell model, 152—154, 179in pure monetary cycle, 165, 258in Tobin model, 122, 124—125

real growth, 12, 174Goodwin model of, 174Keynes—Metzler model of, 318Keynesian, 190—191Solow model of, 13, 24—25, 43, 252, 273

real oscillator, 170real rate of returnon capital, 104on government bonds, 132

real wages, 325, 327—328and Goodwin cycle, 121, 140—141, 143,146, 174, 273

and Keynes—Metzler models, 304,306—307, 315—316, 322

and Keynes—Wicksell models, 38,127—128, 139, 158, 165, 169, 184

and Keynesian dynamics, 191, 193, 197,265, 267, 273

Phillips curve, 36, 46, 127, 273, see alsoPhillips curve

and pure monetary cycle, 159, 161, 165,217—220

and Rose cycle, 34—35, 37—38, 99,

406 Subject index

147—149, 194, 199, 202and Tobin models, 72, 85, 93, 95, 117, 291

relaxation cycle, 56, 90relaxation oscillations, see limit limit cyclesrepressed inflation, see inflation, repressedRosecase under smooth factor substitution,255—258, see also smooth factorsubstitution

effect, 99, 102, 169, 190, 202—203, 207, 293,318, 338

employment cycle, 34—35, 38, 69, 93, 95,127, 146—154, 199, 288, 290, 307

limit cycle, 129, 148, 151, 153, 154, 207model, 32—33, 35—37, 127, 157, 375

Routh—Hurwitz condition(s), 67—68, 99,157—158, 211, 248, 259, 268, 281, 290,292, 302, 318—319

Routh—Hurwitz theorem, 66, 129, 247

saddlepath, 31, 43, 45, 126, 246, 309instability, 29, 31of jump-variable technique, 13stability, 31, 245

saddlepoint, 192, 207, 321—322instability, 28, 47speeds of adjustment of adaptiveexpectations, 42, 47

Sargentanalysis of Cagan framework, 162analysis of Classical model, 40—45, 260analysis of jump-variable technique,46—47, 81, 91—92, 162

analysis of Keynes—Wicksell model, 178,285

analysis of Keynesian dynamics, 136,163—164, 229, 243, 270, 272

analysis of Tobin model(s), 104, 132model, 273

savingsand Classical model, 69, 72function, 25, 33, 37, 142and Goodwin model, 142government, 74, 76, 105and investment, 124, 135, 148—149, 151,185, 189

and Keynes—Wicksell model, 14—15, 31,33, 131, 133—134

private, 25—26, 74, 133propensity, 72, 199rate of, 125, 252ratios, 250and Rose model, 34and Tobin models, 25—26, 37, 74

Say’s Law, 13—14, 25, 27, 30—31, 37, 44, 69,

178on Tobin model(s), 72, 74, 76, 83—84,93—94, 104, 106, 113

self-reference, 341share, market, 261, 326share, wage, 72, 187, 190, 365short run, 16—17, 64, 76—77, 79, 162—163,

181, 189, 216, 275, 336, 338, 375, 380sluggishnessof income adjustments, 199of money wage adjustments, 44, 61,333—334

of output and employment adjustments,276, 341, 345, 354

of price adjustments, 82of price and quantity adjustments, 10, 64,275, 278, 281, 283, 287, 314, 341

of wage adjustments, 92, 159, 273—274,322

of wage and price adjustments, 17, 19—20,22, 127, 212, 276, 278, 282, 299, 314,375—376

smooth factor substitution, 242—277,336—337

and Benassy model, 231and Goodwin model, 36and Keynes—Wicksell model, 127—128and Keynesian growth model, 19, 44, 176,178, 217, 229—231, 278, 376—377

and Rose model, 32, 34—35, 146—148and Tobin models, 80, 98, 102

Solow model, 12—13, 26—28, 32—33, 46, 146,180, 252

stabilityasymptotic (local), 66, 86—87, 99, 102, 120,156, 174, 194, 200, 207, 212—213, 218,249—250, 268, 301—302, 316, 337

and Benassy model, 232, 234Cagan, 87, 121center-type, 97corridor, 287, 301, 303—304, 312global, 32, 37, 66, 81, 98, 148, 154, 183,195, 198, 232, 234, 255, 258, 290, 311

and Goodwin model, 97, 143, 148, 154and Kaldon—Tobin model, 350-353, 357and Keynes—Metzler model, 62, 300—313and Keynes—Wicksell models, 32, 35, 37,135, 137, 139, 253, 255

and Keynesian model, 45, 483, 191,195—199, 200—202, 206, 210—213, 216,266—268

local, 29, 37, 66, 81, 160, 183, 232, 267,290, 322, 357

and pure monetary cycle, 159—162and real and monetary cycle, 227—229

407Subject index

saddlepath, 31, 245and Tobin models, 14, 87, 29, 31, 71,80—82, 86—87, 99—102, 115—119,249—253, 352

stable corridor, see corridor of stabilitystable limit cycle, 86, 91, 250, 169, 301,

303—304, 309, 312, 320, 322stable manifold, 47, 51, 53stable node, 191—192static markup theory of the price level, 180statically endogenous variables, 41, 175,

265, 269, 273, 347stochastic elements, 381stochastic shocks, 381steady stateequilibrium, 15, 60inflation, 38, 309, 330—332, 359, 361rate of interest, 190rate of profit, 117, 139

stockequilibrium, 106, 337see also capital stock

subcritical Hopf bifurcationand Cagan and Tobin stability condition,87


and Keynes—Metzler model, 301substitutionasset demand, 102imperfect, 104, 380neoclassical factor, 15, 326perfect, 106smooth factor, see smooth factorsubstitution

supercritical Hopf bifurcationand Cagan and Tobin stability condition,87


and Keynes—Metzler model, 301—304monetary model, 309real wage and accumulation dynamics,306

superneutrality (of money), 28, 30, 71, 80,103, 251

non-, 69, 97, 103, 251—252supply side, 41, 64, 163, 165, 274Keynesian model, 215, 258, 269, 274, 338,378

Keynesianism, 217, 272shock, 153

taxes, 26, 72—75, 103, 105, 111, 132—133, 377lump sum, 128, 136, 183, 298

net of interest, 136, 181per capital, 286, 297

technical change, 235, 278, 336Harrod—neutral, 239, 241, 377

technology, 10, 230, 260, 268, 316, 319, 347fixed proportions, 74, 128, 147, 235, 237,261, 268, 275, 316

nested CES, 337production, 242, 256, 260, 262, 337

temporary equilibrium, 73, 77, 79, 81, 107,174—175, 184, 269, 272, 273, 377

time rate of change, 347, 269of real wages, 35, 168

Tobin effect, 13, 28, 30, 69, 80, 102, 118enlarged, 251—253

Tobin model(s), 32, 35, 72—73, 76—77, 230,252

bond market extension of, 102—112, 246general, 14, 30, 36, 70, 114, 135, 242general disequilibrium version of,112—123

Kaldor—, see Kaldor—Tobin model(s)and Keynes—Wicksell model(s), 129,135—136

money market disequilibrium extensionof, 82—92

prototype model, 129—136real/money cycle in, 122—124

trend growth, 286, 297, 341—342, 376trend growth rate, 347, 355of capital stock, 342, 365of investment, 356, 357, 361of labor productivity, 236—237, 241

trend term in investment, 36, 39, 132, 236,280, 340—343, 377

ultra short run, 21, 164, 272goods market disequilibria, 195stability, 278

under-unemployment equilibrium, 329underutilized labor and capital, 39, 171, 175,

179, 242, 260, 377unemployment, 229, 315, 329, 342, 377—378classical, 20, 181NAIRU rate of, 329natural rate of, 143, 144, 180, 342steady state of, 280, 340Tobin—Keynesian cycle, 329

unit of capital, 33, 128, 132, 155, 181unit(s)equity, 132money, 132wage, 273

unstable limit cycle, 169, 250, 301, 320unstable node or focus, 191, 193

408 Subject index

utilizationcapacity, see capacity utilizationexpectations, 232rate of capital and labor, 18, 24, 62,179—180, 274, 316, 376

variablesdynamic, 34, 43, 45, 118, 254, 286, 308expectational, 48, 54state, 79, 85, 11, 273, 315, 319, 351, 356statically endogenous, 41, 175, 273, 347

viability, 229, 282, 358, 375of Benassy model, 234of Kaldor—Tobin (endogenous long-runemployment and growth) model, 385,371—372

of Keynes—Metzler model, 315, 334of real cycle, 205, 207of real and monetary cycle, 169of Rose employment cycle, 149, 257—258of Tobin type models, 91, 123

wage adjustment, 191, 198, 314, 321—322,329, 332, 381

Benassy’s, 375

equation, 94function, 194money-, 44, 3 34nonlinearity in, 202—204, 282, 319

wage flexibility, 15, 33, 94, 122, 204, 211, 234and Goodwin model, 142and Keynes—Metzler model(s), 306—307,313, 322

and Keynes—Wicksell model, 33—34, 257money-, 35, 158

wage—price spiral, 19, 62, 335wage rigidity, 163, 329money-, 272

wage share, 72, 187, 190, 365Walras’ Law, 27, 30—31, 37, 75, 133wealth, 21, 65, 106, 130, 132, 377allocation of, 54—55constraint, 88, 105, 106, 135net, 110, 112real, 72, 104—105, 132

Wickselliandemand-pressure price-inflation, 14price dynamics, 16, 123theory of inflation, 15, 164, 272

working model, see Keynes—Metzler model

409Subject index

the dynamics of keynesian monetary growth-macrofoundations-chiarella & flaschel

Documents